Subject Class Calendar Fall 2009

Date Day Lesson HW Complete

Lesson #1 Aim: How do we perform operations involving monomials and polynomials?

Yes

Lesson #2 Aim: How do we divide polynomials? YesLesson #3 Aim: How do we solve first degree inequalities

Yes

Lesson #4 Aim: How do we solve compound linear inequalities ?

Yes

Lesson #5 Aim: How do we graph absolute value relations and functions?

Yes

Lesson #6 Aim: How do we solve equations involving absolute values?

Yes

Lesson #7 Aim: How do we solve absolute value inequalities?

Partial

Lesson #8 Aim: How can we factor polynomials? Yes

Lesson #9A Aim: How do we factor the difference of two perfect squares or polynomials completely?

Yes

Lesson #9B Aim: How do we solve quadratic equations by factoring?

Partial

Lesson # 10 Aim: How do we graph the parabola y = ax 2 + bx + c?

Partial

Lesson #11 Aim: How do we solve and graph a quadratic inequality?

Partial

Lesson #12 Aim: How can we use the graph of a parabola to solve quadratic inequalities in two variables?Lesson #13 Aim: How do we solve more complex quadratic inequalities?Lesson #14 Aim: How do we simplify radicals? PartialLesson #15 Aim: How do we add and subtract radicals?

Partial

Lesson #16 Aim: How do we multiply and divide radicals?

Partial

Lesson #17 Aim: How do we rationalize a fraction with a radical denominator (monomial or binomial surd)?

Partial

Lesson # 18 Aim: How do we complete the square? PartialLesson #19 Aim: How do we apply the quadratic formula to solve quadratic equations with rational roots?Lesson #20 Aim: How do we apply the quadratic formula to solve quadratic equations with irrational roots?

Partial

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Lesson #21 Aim: How do we apply the quadratic formula to solve verbal problems?Lesson #22 Aim: What are complex numbers and their properties?

Partial

Lesson #23 Aim: How do we add and subtract complex numbers?

Partial

Lesson #24 Aim: How do we multiply complex numbers?

Partial

Lesson #25 Aim: How do we divide complex numbers and simplify fractions with complex denominators?

Partial

Lesson #26 Aim: How do we find complex roots a quadratic equation using the quadratic formula?

Partial

Lesson 27: Aim: How do we use the discriminant to determine the nature of the roots of a quadratic equation?Lesson #28 Aim: How do we find the sum and product of the roots of a quadratic equation?

Partial

Lesson #29 Aim: How do we solve systems of equations using the graphing calculator? Lesson #30 Aim: How do we solve systems of equations

Partial

Lesson #31 Aim: How do we reduce rational expressions?

Partial

Lesson #32 Aim: How do we multiply and divide rational expressions?

Partial

Lesson #33 Aim: How do we add and subtract rational expressions with like or unlike denominators?

Partial

Lesson #34 Aim: How do we add and subtract rational expressions with unlike polynomial denominators?Lesson #35 Aim: How do we reduce complex fractions?

Partial

Lesson #36 Aim: How do we solve rational equations? Lesson #37 Aim: How do we solve rational inequalities?Lesson #38 Aim: How do we evaluate expressions involving negative and rational exponents?

Partial

Lesson #39 Aim: How do we find the solution set for radical equations?

Partial

Lesson #40 Aim: How do we find the solution set of an equation with fractional exponents?

Lesson #41 Aim: What are relations? Partial

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Lesson #42 Aim: What are functions? PartialLesson #43 Aim: How do we use function notation? PartialLesson #44 Aim: What are compositions of functions?

Partial

Lesson #45 Aim: How do we find the inverse of a given relation?

Partial

Lesson #46 Aim: What is an exponential function? PartialLesson #47 Aim: What is the inverse of the exponential function?

Partial

Lesson #48 Aim: How do we find the log b a? Lesson #49 Aim: How do we use logarithms to find values of products and quotients?Lesson #50 Aim: How do we use logarithms for raising a number to a power or finding roots of numbers?Lesson #51 Aim: How do we solve exponential equations?Lesson #52 Aim: How do we solve exponential and logarithmic equations?

Lesson #53 Aim: How do we solve verbal problems leading to exponential or logarithmic equations?Lesson #54 Aim: What is a transformation? How do we do Transformations involving reflections?

Partial

Lesson 55 Aim: What are geometric translations, dilations and rotations?

Partial

Lesson #56 Aim: How do we perform transformations of the plane on relations and functions?Lesson #57 Aim: How do we graph and write the equation of a circle?

Partial

Lesson #58 Aim: What is direct and inverse variation? PartialLesson #59 Aim: How do we find the roots of polynomial equations of higher degree by factoring and by applying the quadratic formula?

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Lesson #1 Aim: How do we perform operations with polynomial expressions containing rational coefficients?Students will be able to1. Add and subtract polynomials with rational coefficients2. Multiply and divide monomials with exponents and rational coefficients3. Simplify parenthetical expressions (nested groupings)4. Multiply polynomials with integral and rational coefficients5. Explain the procedures used to add, subtract, multiply and divide monomials and polynomials

Homework:

Do Now: Simplify3xy+2xy+10xy=15xy 9ab(2ab3)=18a2b4 (x+5)(x+4)=x2+9x+20

Definitions:Monomial -An expression which contains only one term, for instance (2a,3xy,5, x2)Binomial - A polynomial of two terms such as 10a+12b is a binomial What is a bigomist? Man with 2 wives. What is a bisexual? A person who goes with 2 different sexes.Trinomial - A polynomial of three terms such as x2+3x+2Polynomial- is a sum of monomials. What is a polygon? A polygamist?

Coefficient- If a terms has a variable, the numerical factor is called the coefficient Example –1x+5Y+17; -1X and 5Y are terms; -1 and 5 are coefficients, 17 is a

constant term. Term- An algebraic expression that is a number, a variable or the product or quotient of numbers or variables. Not separated by a + or a -.

Example 5, 2x, -4/7y2, .7a2b3 , 5x+5, 3xy, 5ab,7x2

Example of separate terms: 2x+y, 3x2+y, 2x+5

Like Terms- Two or more terms that contain the same variable, with corresponding variables having the same exponents. Example: 6k and k; 5x2 and –7x2; 9ab and 2/5ab

Unlike Terms- Two or more terms that contain different variables, or the same variables with different exponents. Example 3x and 4y; 5x2 and 5x3; 2xy and 2x

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Standard Form- The terms decrease in terms from left to right.

Example  Add:  (2x2 - 4) + (x2 + 3x - 3)

1. Using a horizontal method to add like terms: Remove parentheses.   Identify like terms.  Group the like terms together.Add the like terms.(2x2 - 4) + (x2 + 3x - 3) = 2x2 - 4 + x2 + 3x – 3 = 2x2 + x2 + 3x - 4 – 3 = 3x2 + 3x – 7

Using a vertical method to add like terms:Arrange the like terms so that they are lined up under one another in vertical columns.  Add the like terms in each column following the rules for adding signed numbers.2x2 + 0x - 4+ x2  + 3x - 3---------------------3x2 + 3x - 7

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To add polynomials, combine like terms by adding their numerical coefficients. To subtract a polynomial add the opposite (additive inverse) (p9)

Examples

Classwork and Homework

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Lesson #2 Aim: How do we divide polynomials?Students will be able to1. Divide a polynomial by a polynomial including polynomials with rational coefficients; with and without remainders2. Apply the operations of multiplication and addition of polynomials to check quotients

Do Now:

Homework:

Steps for Dividing a Polynomial by a Monomial: 1. Divide each term of the polynomial by the monomial.

a)  Divide numbers (coefficients)

b)  Subtract exponents                 2.  Remember that numbers do not cancel and disappear!  A number divided by itself is 1.  It reduces to the number 1.

3.  Remember to write the appropriate sign in between the terms.

Notice how the numbers (the coefficients) were divided. 

The polynomial on the top has 3 terms and the answer has 3 terms.

Answer:

 

Notice how the exponents were subtracted.

Notice how the last term reduced to one.

Steps for Dividing a Polynomial by a Binomial:

1. Remember that the terms in a binomial cannot be separated from one another when reducing.  For example, in the binomial 2x + 3, the 2x can never be reduced unless the entire expression  2x + 3 is reduced.

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2. Factor completely both the numerator and denominator before reducing.3. Divide both the numerator and denominator by their greatest common factor.

Examples: 

1.

 

How do we factor a binomial?

2.

 

 How do we factor the difference of 2 perfect squares?

3.

 

 How do we factor a trinomial?

4.

 

Notice that the -1 was factored out of the numerator to create a binomial compatible with the one in the denominator.2 - x = -1(x - 2)

Practice:

 

 

Lesson #3 Aim: How do we solve first degree equations and inequalities?

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Students will be able to1. Apply the postulates of equality to solve first degree equations algebraically2. Apply the postulates of inequality to solve first degree inequalities algebraically3. Solve first degree equations and inequalities graphically4. Graphically justify the solution of each linear equation and inequality found algebraically

Do Now:

Homework:

Vocabulary:An equation- is a sentence that uses the symbol = to state that two quantities are equal. An inequality- is a sentence that uses one of the symbols of order, namely >,<, , .Solution Sets- Finite, infinite or empty. Subset of the domain that make the sentence true.Open Sentence- One or more of the quantities in the relationship contains a variable. The sentence is neither true nor false until the variable is replaced with one of the elements of the domain.Closed Sentence- Or a statement that can be judged to be true or false. No variables.

Properties of Equality:Add/Subtract Property

For all numbers a, b, & c, if a = b, then a + c = b + c and a - c = b - c.

Mult/Division Property

For all numbers a, b, and c, if a = b, then a * c = b * c, and if c not equal to zero, a ÷ c = b ÷ c.

Properties of InequalityAdd/Subtract Property If equals are added or subtracted from unequals the sums or

differences are unequal in the same order.Mult/Division Property

If unequals are multiplied or divided by positive equals, the products or quotients are unequal in the same order. If they are multiplied by negative equals the products or quotients are unequal in the opposite order.

Distributive Property For all numbers a, b, & c, a(b + c) = ab + ac.

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Model Example Solve and check for an equality:5(x-1)-3=2x-(3-x) 5x-5-3=2x-3+x Simplify Each Side5x-8=3x-3 Use additive inverse to have variable on one side2x=5 Use multiplicative inversex=5/2

Sample problems:

-3x+9=24-3x=15, x=-5X=12

K=-39

Solving InequalitiesModel example for an inequality2(5-x)>3+1 10-2x>4 -2x>-6 x<3Steps: Clear parentheses, combine like terms, use additive inverse, use multiplicative inverse, change direction of inequality.

Graphing inequalties on a number line.

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Lesson #4 Aim: How do we solve and graph compound linear inequalities involving the conjunction and disjunction?Students will be able to1.Graph inequalities on a number line2. Solve and graph conjunctions and disjunctions of two inequalities3. Apply graphing compound inequalities to problems involving the conjunction or the disjunction

Do Now: What is the definition of a compound sentence in English? What is a compound in science? What is a compound equality in math? What is a compound inequality in math?

Homework:

VocabularyCompound Inequality- Two inequalities that are joined by the word and or the word or. The solution of one joined by and is any number that makes both inequalities true. Two inequalities joined by or is any number that makes either inequality true.

A compound linear inequality is one that has two inequalities in one problem.  For example, 5 < x + 3 < 10 or -1 < 3x < 5.  You solve them exactly the same way you solve the linear inequalities shown above, except you do the steps to three "sides" (or parts) instead of only two.

 Example 9:  Solve, write your answer in interval notation and graph the solution set:   This is an example of a compound inequality 

Interval notation:

Graph:

*Inv. of add. 2 is sub. by 2 *Apply steps to all three parts     

*All values between -6 and 8, with a closed interval at -6 (including -6)

*Visual showing all numbers between -6 and 8, including -6 on the number line. 

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Interval notation: This time we have a mixed interval since we are including where it is equal to -6, but not equal to 8.   x is between -6 and 8, including -6, so -6 is our smallest value of the interval so it goes on the left and 8 goes on the right.   The boxed end on -6 indicates a closed interval on that side. A curved end on 8 indicates an open interval on that side.  Graph :   Again, we use the same type of notation on the endpoints as we did in the interval notation, a boxed end on the left and a curved end on the right.  Since we needed to indicate all values between -6 and 8, including -6, the part of the number line that is in between -6 and 8 was darkened.  Compound Inequalities – Conjunctions

To solve a compound inequality, you must solve each part separately. A compound inequality containing and is true only if both parts of it are true. This means, the graph of a compound inequality containing and must be the

intersection of the graphs of the two solution parts. Where the graphs overlap or intersect determines the solution set.

Example 1

7<2x+5<177<2x+5 2x+5<177-5<2x 2x<121<x x<6

Example 2 -2x-6>4 or x+5>4-2x>10 or x>-1x<-5

solution set is x<-5 or x>-1

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Model Example 1

Model Example 3 4v+3<-5 or 2v+7<1

Collaborative Group Activity

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Lesson #5 Aim: How do we graph absolute value relations and functions?Students will be able to1. Explain how to use the graphing calculator to graph absolute value relations and functions2. Determine the appropriate window for each graph3. Use the graphing calculator to graph absolute value relations and functions4. Apply the transformations f(x+a), f(x) +a, -f(x), and af(x) to the absolute value function

Need Graphing Calculators and OverheadDo Now:Describe what the graph of the following 2 equations would look like.y=-2x+3 Straight line, negative slope of -2, y intercept of 3.y=x2+2 Parabola, Opens upward

Homework:

Complete the following table: Describe each graph using math terminology.Type Equation DescriptionLinear Functions

Y=mx+b or f(x) =mx+b

Graph is a straight line with m = slope and b= y intercept

y=-4x-3 Slope = -4 and y intercept =-3

Constant Function

y=b or f(x) =b Every x corresponds to a constant function y.

y=3

Quadratic Function

y=ax2+bx+c f(x)= ax2+bx+c

Graph is a parabola whose axis of symmetry is a vertical line. Every x value corresponds to one and only one y value the relation is a function

y=x2-4x+3

Absolute Value

Y=|x| for every value of x there is one and only one value of y. A V shape

Y=|2x-4|

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Function Graphing Absolute Value Equations

Example 1:     Solve:   Enter left side in Y1.  You can find abs( ) quickly under the CATALOG (above 0)( or MATH → NUM, #1 abs( )Enter right side in Y2Use the Intersect Option (2nd CALC #5) to find where the graphs intersect.  Move the spider near the point of intersection, press ENTER.  Simply hit ENTER twice more.  You must repeat this process to find the second point of intersection.Answer:  x = 4;  x = -4

                                                   

Example 2:    Solve:  Answer:  x = 2;  x =

3.3333333The x value is stored in the calculator's memory.  If you wish to change 3.333333333 to a fraction, simply return to the home screen, hit x, hit Enter.  Now, change to fraction. (MATH, #1►Frac ) Answer:  x = 2;  x = 10/3

 

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Lesson #6 Aim: How do we solve linear equations in one variable involving absolute values?Students will be able to1. State the definition of the absolute value of x2. Apply the definition of absolute value to solve linear equations involving absolute values3. Graph solutions to linear absolute value equations on the number line4. Verify solutions to absolute value equations5. Solve verbal problems resulting in linear equations involving absolute value

Do Now: What is the definition of absolute value? Can the absolute number of a negative number ever be negative?

Homework:

To solve an absolute value equation, isolate the absolute value on one side of the equal sign, and establish two cases:

Case 1:               Set the expression inside the             absolute value symbol equal to             the other given expression.

Case 2:               Set the expression inside the             absolute value symbol equal to             the negation of the other given             expression

 . . .  and always CHECK your answers.The two cases create "derived" equations.  These derived equations may not always be true equivalents to the original equation.  Consequently, the roots of the derived equations MUST BE CHECKED in the original equation so that you do not list extraneous roots as answers.

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Review the following procedures

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Lesson #7 Aim: How do we solve linear absolute value inequalities involving one variable? (pg 81-82 Amsco)Students will be able to1. Solve inequalities of the form |x| < k and |x| > k2. Solve inequalities of the form x ± a < k and x ± a > k3. Graph solutions to linear absolute value inequalities on the number line4. Verify solutions to absolute value inequalities5. Solve verbal problems resulting in linear inequalities involving absolute valueDo Now:

3z=3 3z=-3 z=1 z=-1

Homework:

Procedures Solving an absolute value inequality problem is similar to solving an absolute value

equation.  Isolate the absolute value on one side of the inequality symbol, then follow the rules

below:

If the symbol is > (or >) :          (or) If  a < 0, all real numbers will satisfy .

If  a > 0, then the solutions to are  x > a  or  x < - a.

Think about it:   absolute value is always positive (or zero), so, of course,  it is greater than any negative number.

If the symbol is < (or <) :          (and)

If  a > 0, then the solutions to are  x < a  and  x > - a.Also written:   - a < x < a.

If  a < 0, there is no solution to .

Think about it:  absolute value is always positive (or zero), so, of course,  it cannot be less than a negative number.

 

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R E M E M B E R: When working with any absolute value inequality, you must create two cases.  To set up the two cases:

If <, the connecting word is "and".   If >, the connecting word is "or".x < aCase 1:  Write the problem without the absolute value sign, and solve the inequality.

x > -aCase 2:  Write the problem without the absolute value sign, reverse the inequality, negate the value NOT under the absolute value, and solve the inequality

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Model Examples|2x+3|<7-7<2x+3<7 Write a derived equation-10<2x<4 Solve-5<x<2Graph the solution -6 -5 -4 -3 -2 -1 0 1 2 3

|3+y|-2 ≥0|3+y|≥23+y≥2 3+y≤-2 Write derived equationsY≥-1 y≤-5Graph the solution -6 -5 -4 -3 -2 -1 0 1 2 3

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Lesson #8 Aim: How do we factor polynomials? (Page 22 Amsco)Students will be able to1. Recognize when to factor out a greatest common factor2. Factor by extracting the greatest common factor3. Identify and factor quadratic trinomials, where a > 14. Factor quadratic trinomials that are perfect squares5. Justify the procedures used to factor given polynomial expressionsDo Now:

Factor 120 to lowest prime factors 15*4, (5*3*2*2)

2) Factor 48x3 3*2*2*2*2*x*x*x

3) Factor 15x2+30x 15x(x+2) 4) Factor x2-49

Homework:

Example 1

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Example 2

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Example of factoring trinomials

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Factoring Trinomials with coefficients >1

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Lesson #9A Aim: How do we factor the difference of two perfect squares and factor polynomials completely?Students will be able to1. Recognize and factor the difference of two perfect squares2. Compare factoring the difference of two perfect squares to factoring a trinomial3. Explain what is meant by factoring completely4. Factor polynomials completely5. Justify the procedures used to factor polynomials completely6. Explain why it is more efficient to factor completely by first extracting the GCF7. Factor cubic expressions of the form a3 - b3 or a3 + b3 (enrichment only)

Do Now: Factor Completely1) 3x2-12x= 3x(x-4) 2) 9y3+3y2= 3y2(3y+1) 3) 4x2+4x+1= (2x+1)(2x+1)

Homework:

What is a perfect square? Have students list perfect squares. What does it mean to factor completely?

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Lesson #9B Aim: How do we solve quadratic equations by factoring?Students will be able to1. Justify and explain the multiplication property of zero2. Explain what is meant by the standard form of a quadratic equation3. Transform a quadratic equation into standard form4. Factor the resulting quadratic expression5. Apply the multiplication property of zero to solving quadratic equations6. Check the answers in the original equation

Do Now:

Homework:

Quadratic equations are normally expressed as  , where a does not equal zero. Many of the simpler quadratic equations with rational roots can be solved by factoring.

To solve a quadratic equation by factoring: 1 Start with the equation in the form

Be sure it is set equal to zero!2 Factor the left hand side (assuming zero is on the right)3 Set each factor equal to zero4 Solve to determine the roots (the values of x)

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Some possible situations:

Factoring with GCF(greatest common factor)

Factoring with DOTS(difference of two squares) Factoring Trinomials

Find the largest value that can be factored from each of the elements of the expression.

Look carefully at this example to refresh this

process:

In a quadratic equation in descending order with a leading coefficient of one, look for the product of the roots to be the constant tern and the sum of the roots to be the coefficient of the middle term.

 

Or Isolate the Variable

Factoring Harder Trinomials  Tricky One!!If the leading coefficient is not equal to 1, you must think more carefully

about how to set up your factors.

 

Be sure to get the equation set equal to zero before you factor.

Graph  2x2 - 5x + 2

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Quadratic equations are of the form where a, b and c are real numbers and ..  Quadratic equations have two solutions.  It is possible that one solution may repeat.

Solving by FactoringExample 1:  Solve by factoring:

Example 2:Solve by factoring:

Example 2:Solve by factoring:

Solve by Graphing

Example 1:  Solve by graphing:

     

Method 1:  Set the equation equal to zero, if necessary.  Find the roots using the ZERO

command tool of the graphing calculator.  For help with the calculator, click here.

Example 2:  Solve by graphing:

    

Method 2:  Graph each side of the equation separately.  Use the INTERSECT command

tool to find when the graphs cross.  Repeat this process for both intersection points.  For help

with the calculator, click here.

 

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Solving by Quadratic FormulaThe solutions of some quadratic equations are not rational, and cannot be factored.  For such equations, the most common method of solution is the quadratic formula.  The quadratic formula can be used to solve ANY quadratic equation, even those that can be factored. Be sure you know this formula!!! Note:  The equation must be set equal to zero before using the formula.Example: 

 

 

As decimal values:

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Lesson # 10 Aim: How do we graph the parabola y = ax2 + bx + c?Students will be able to1. create a table of values and graph a parabola2. explain the effect of a, b, and c on the graph of the parabola3. identify the key elements of a parabola (axis of symmetry, turning point, intercepts, opening up/down)4. investigate and discover the effects of the transformations f(x+a), f(x) + a, af(x), f(-x) and –f(x) on the graph of a parabola5. produce a complete graph of a parabola using a graphing calculator6. use the graphing calculator to identify the roots of a quadratic equationWriting Exercise: Video games use life-like graphics to model 3-D effects on the 2-D computer monitor. Video golf games show the path of a golf ball that is hit by each player. Describe the shape of the path of the ball as shown on the screen and speculate on the type of equation(s) the programmer needed to use in order to produce this graphic.

Homework: Graph the following equations by creating a table of valuesY=5x2 Y=1/2x2 y=-7x2

Development:/Motivation/Do Now:Solve graphically: y = 2x + 1 by creating a table of values and graphingx 2x+1 y-2 2(-2)+1 -3-1 2(-1)+1 -10 2(0)+1 11 2(1)+1 32 2(2)+1 5Pivotal QuestionsWhat were the steps we followed to create the graph?

Create a table of values (What goes in each column? How many rows?) Select values of x to substitute in the equation (How many values do we need? What

values of x should we use?) Solve for y Label x and y axis, Put in a scale for both axis Plot ordered pairs of x and y Connect the Points

What are the characteristics of this graph? (Linear, Straight Line, Slant Upwards)Is it a positive or negative slope? (Positive)Why? (As we go left to right y goes up.)What is the slope of the line? (2) How did you calculate the slope? (y2-y1/x2-x1)(or Change in y over the change in x)

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Ok I want everyone to work with me in graphing a different type of equation

What are the steps we are going to follow to graph this type of equation? Create a Table of Values Select values of x to substitute ( How many values do we need?) Solve for y (Do we multiply by 2 first or square the value of x?) Label x and y axis, Put in a scale for both axis Plot the x,y values on a graph Connect the points

Before we all sketch the graph lets look at the values of y and xIs there any pattern to the numbers. Look at –2 and +2 (y has the same value)As x increases by 1 what happens to the value of y? Does it change by a constant amount? (No)

Lets all of us sketch the graph on the same chart that we used for a linear equation.How would you describe the graph? (U shaped)What happens as we move upward? (The opening gets wider)What would happen if we fold the graph in half (Symmetry)Where have we seen shapes like this in the real world.

Satellite dish; Parabolic microphone used at football games; Solar Hot Dog cooker; clothes line; archway; path of a basketball being shot, baseball being hit to the outfield, football being kicked.

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Y=3x2

Similarities DifferencesLinear Linear QuadraticCan be graphed on Coordinate Plane

Straight Line Curved Line

Same procedure to create graph No variable to the second power.

One of variables is to the second power (squared)

Lines go on forever Constant Slope Different slopeSolid line Turning Point

Need only 2 points to graph

Need more than 2 points to graphAxis of Symmetry

Vocabulary:Vertex- Highest or lowest point on a parabola.Axis of symmetry- The line down the middle of the parabola.Quadratic- it contains the variable squared, but not raised to any higher power. For instance a quadratic equation in x contains x2 but not x3

Quadratic Equation- An equation which asks us to find where a particular quadratic polynomial is equal to zero. It can always be written in the form: ax2+bx+c = 0. Parabola - A particular shape of curve, given by an equation of the form y=ax2+bx+c (a, b, c constants, a nonzero). The path of a thrown object is a parabola.

Group Activity- Graph the following equations: Y=2x2+1 Y = x2 - 4 y = x2 – 2x+2x 2x2+1 y x X2-4 y x x2 + 2x+2 y-3 2(-3)2+1 19 -3 (-3)2-4 5 -3 -32+2(-3)+2 5-2 2(-2)2+1 9 -2 (-2)2-4 0 -2 -22+2(-2)+2 2-1 2(-1)2+1 3 -1 (-1)2-4 -3 -1 -12+2(-1)+2 10 2(-0)2+1 1 0 (-0)2-4 -4 0 -02+2(0)+2 21 2(1)2+1 3 1 (1)2-4 -3 1 12+2(1)+2 52 2(2)2+1 9 2 (2)2-4 0 2 22+2(2)+2 103 2(3)2+1 19 3 (3)2-4 5 3 32+2(3)+2 17

Summary: How are these equations different? How are they the same? How do we graph these type of equations? How many points do we need? What values should we use? What is the shape of the graph of a linear equation? What is the shape of the graph of a quadratic equation? Where do we see shapes like that in real life?

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Lesson #11 Aim: How do we solve and graph a quadratic inequality algebraically?Students will be able to1. transform a quadratic inequality into standard form2. solve a quadratic inequality algebraically and graph the solution on a number line3. write the solution to a quadratic inequality as a compound inequalityWriting Exercise: How can the graphing calculator be used to verify the solution set of its related quadratic inequality?

Do Now:

Homework:

A quadratic inequality is one that can be written in one of the following standard forms:

or or or Quadratic Inequalities involve variable expressions that look just like the expressions found in quadratic equations.  The only difference is an inequality symbol ( ,  ,  ,  ) instead of an equals sign. For example:

or

The key is to make sure you have a 0 on one side of the inequality before you factor.  We will then use a "special number line" to keep track of our work. For example, to solve the inequality

 Step 1: Make sure you have a 0 on one side of the inequality.

 Step 2: Factor (if possible).

Our goal is to determine when this product is positive or equal to zero.  Step 3: Construct a number line and identify the x values which make the individual factors equal to 0.  x = 0 and x = 1 will make the factors equal zero. So we have the number line

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Step 4:  Use the number line and the factored form to determine where the product is positive or equal to zero. We've already seen that the product equals zero when x = 0 and when x = 1.  So we designate these two values as part of our solution by "shading them in" on the number line.

Our number line is now divided into three pieces.  all numbers less than 0 all numbers between 0 and 1 all numbers greater than 1

Basic Strategy: Working with one piece at a time, we can choose a test value from each of these pieces and determine whether that test value causes our product to be positive or negative. In this example our product consists of the factored form Let's try an x value less than 0, say x = - 1.  Plugging in x = - 1 yields ( - 3)( - 2) = 6, a positive number. So our product is positive when x < 0.

Let's try an x value between 0 and 1, say x =0 .5.  Plugging in x = .5 yields (1.5)( - 0.5) = -0 .75, a negative number. So our product is negative when 0 < x < 1.

Let's try an x value greater than 1, say x = 2.  Plugging in x = 2 yields (6)(1) = 6, a positive number. So our product is positive when x > 1.

We designate these results on the number line by "shading in" the pieces which produce the desired outcome. In this example, we shade in the pieces which make the product positive.

So the solution set for the inequality  is { all real numbers x such that x  0 or x  1 }. We can also write this solution using interval notation

Note: If the inequality in this example were "  " instead of "  "  our solution set would not include the endpoints x = 0 and x = 1. Remember, in this method, the factored form will tell you where to "break up the number line," and which pieces to keep, simply by using test numbers to determine the sign of each piece seperately.

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Special Note: If the given quadratic has no real solutions, then the expression will either always be positive or always be negative. In this case, simply plug x = 0 into the quadratic  and the resulting number value will allow you to answer the question.

Special Case: If the given quadratic does not factor conveniently, we must use the quadratic formula to determine which numbers (if any) should be used to break up the number line.Example 1 Solve the given inequality and give your answer using interval notation as well as inequality notation.

Solution Step 1: Multiply out and make sure you have a 0 on one side of the inequality.

We now have a quadratic inequality.  Step 2: Factor (if possible). In this example, our goal is to determine when this product is positive.  Step 3: Construct a number line and identify the x values which make the individual factors equal to 0.  In this case, x = - 4 and x = - 3 will make the factors equal zero. So we have the number line

Step 4:  Use the number line and the factored form to determine where the product is positive. We've already seen that x = - 4 and x = - 3 make the product equal zero. Thus, these values do not yield the desired outcome.   We designate this result with "open circles" on the number line at x = - 4 and x = - 3.

Our number line is now divided into three pieces.  Namely:

all numbers less than – 4, all numbers between - 4 and – 3, all numbers greater than - 3

Basic Strategy: Working with one piece at a time, we can choose a test value from each of

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these pieces and determine whether that test value causes our product to be positive or negative. In this example our product consists of the factored form Let's try an x value less than - 4, say x = - 5.      Plugging in x = - 5 yields ( - 1)( - 2) = 2, a positive number. So our product is positive when x < - 4.

Let's try an x value between - 4 and - 3, say x = - 3.5.      Plugging in x = - 3.5 yields (.5)( - .5) = - 0.25, a negative number. So our product is negative when - 4 < x < - 3.

Let's try an x value greater than - 3, say x = 0.      Plugging in x = 0 yields (4)(3) = 12, a positive number.     So our product is positive when x > - 3.

We designate these results on the number line by "shading in" the pieces which produce the desired outcome. In this example, we shade in the pieces which make the product positive.

So the solution set for the inequality  is { all real numbers x such that x < - 4 or x > - 3 }. We can also write this solution using interval notation

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Lesson #12 Aim: How can we use the graph of a parabola to solve quadratic inequalities in two variables?Students will be able to1. identify the key elements (axis of symmetry, turning point, intercepts, opening up/down) of a parabola2. apply transformations to the graph of a parabola3. produce a complete graph of a parabola using a graphing calculator4. use the graphing calculator to calculate the roots of a quadratic equation5. solve a quadratic inequality in two variables graphically6. write the solution to a quadratic inequality as a compound inequalityWriting Exercise: In class, we plot 6 or 8 points to define the complete shape of a parabola. How do we determine what x-values to use in order to create a complete graph of the parabola?

Lesson #13 Aim: How do we solve more complex quadratic inequalities?Students will be able to1. transform a quadratic inequality into standard form2. solve quadratic inequalities3. graph the solution set of a quadratic inequality on the number line4. apply graphing to the conjunction and the disjunction

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5. graph the related parabola to identify the solution to a quadratic inequality

Lesson #14A Monday November 28Aim: What are Roots and RadicalsObjectives: Students will be able to

Understand the meaning of square roots and cube roots Understand the nth root of a number Evaluate simple expressions involving square roots

Homework: Page 401 # 6-12,19-22Do Now:

Vocabulary:Square Root- is one of two equal factors whose product is that number. Every positive real number has two square roots.Principal Square Root- of a positive number is its positive square root.Cube Root- Is one of 3 equal factors whose product is that number.N th Root - Is one of n equal factors whose product is that number.

where k=radicand, n =the index

The square root of 0 is 0. The square root of a negative number are not real numbers.

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Find the following

The square root of a product of positive numbers is equal to the product of the square roots of the numbers.

=

The simplest form of a square root radical has the greatest perfect square factored out. The square root of a quotient of positive numbers is equal to the quotient of the square

roots of the numbers.

=

=

Lesson #14B Aim: How do we simplify radicals?Objectives: Students will be able to simplify radicals with a numerical index of 2 or 3. simplify radicals involving literal radicands. explain the procedure for simplifying radical expressions. explain how to determine when a radical is in simplest form.

Homework: Day 1 9-4 Worksheet Simplifying Radicals- MultiplicationDay 2 9-4 Worksheet Division with RadicalsDay 3 Page 406 # 5-12,32-36

Do Now:

Procedure To simplify (or reduce) a radical:

Find the largest perfect square which will divide evenly into the number under your radical sign.  This means that when you divide, you get no remainders, no decimals, no fractions.Reduce:

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the largest perfect square that divides evenly into 48 is 16. If the number under your radical cannot be divided evenly by any of the perfect squares, your radical is already in simplest form and cannot be reduced further.

2.  Write the number appearing under your radical as the product (multiplication) of the perfect square and your answer from dividing.

3) Give each number in the product its own radical sign.

4.  Reduce the "perfect" radical which you have now created.

5.  You now have your answer.

What happens if I do not choose the largest perfect square to start the process? 

If instead of choosing 16 as the largest perfect square to start this process, you choose 4, look what happens.....

 Unfortunately, this answer is not in simplest form.  The 12 can also be divided by a perfect square (4).

If you do not choose the largest perfect square to start the process, you will have to repeat the process.  

Example:  Reduce

Don't let the number in front of the radical distract you.  It is simply "along for the ride" and will be multiplied times our final answer. The largest perfect square dividing evenly into 50 is 25.

Reduce the "perfect" radical and multiply times the 3 (who is "along for the ride")

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Lesson #15 Aim: How do we add and subtract radicals?Students will be able1. add and subtract like radicals with numerical or monomial radicands2. add and subtract unlike radicals with numerical or monomial radicands3. express the sum or difference of radicals in simplest form4. explain how to combine radicalsWriting Exercise: Why is x6 a perfect square monomial and x9 not a perfect square monomial?

When adding or subtracting radicals, you must use the same concept as that of adding or subtracting like variables.  

In other words, the radicals must be the same before you add (or subtract) them.

Since the radicals are the same, simply add the numbers in front of the radicals (do NOT add the numbers under the radicals).

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1.  Are the radicals the same? NO2.  Can we simplify either radical? Yes,  can be simplified.

3.  Simplify the radical.

4.  Now the radicals are the same and we can add.

1. Simplify  Answer:2. Simplify Answer:3. Since the radicals in steps 1 and 2 are now the same, we can combine them.

4. You are left with:

5. Can you combine these radicals? Answer:  NO6. Therefore, Answer: ------------------>

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Lesson #16 Aim: How do we multiply and divide radicals?Objectives: Students will be able to: multiply radical expressions, divide radical expressions, express the products and quotients of radicals in simplest form, express fractions with irrational monomial denominators as equivalent fractions with rational denominators.

Do Now:

Homework: When multiplying radicals, one must multiply the numbers OUTSIDE (O) the radicals AND then multiply the numbers INSIDE (I) the radicals.

When dividing radicals, one must divide the numbers OUTSIDE (O) the radicals AND then divide the numbers INSIDE (I) the radicals.

1.  Multiply the outside numbers first 2 • 3 = 6 2.  Multiply the inside numbers3.  Put steps 1 and 2 together and simplify4.  Therefore, the answer is 72.

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1.  Divide the outside numbers first.

2.  Divide the inside numbers.

3.  Put steps 1 and 2 together and simplify.4.  Therefore, the answer is:

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Lesson #17 Aim: How do we rationalize a fraction with a radical denominator?Objectives: Students will be able to: rationalize monomial denominators, rationalize binomial denominators, express results in simplest form.

A fraction that contains a radical in its denominator can be written as an equivalent fraction with a rational denominator. Never leave a radical in the denominator of a fraction. Always rationalize the denominator.

1.  When the denominator is a monomial (one term), multiply both the numerator and the denominator by whatever makes the denominator an expression that can be simplified so that it no longer contains a radical. Sometimes the value being multiplied happens to be exactly the same as the

denominator, as in Example 1:Multiplying the top and bottom by will create the smallest perfect square under the square root in the denominator. Replacing by 7 rationalizes the denominator.

Sometimes you need to multiply by whatever makes the denominator a perfect square or perfect cube or any other power that can be simplified, as in Examples 2 and 3.

 Multiply by a value that will create the smallest perfect square under the radical.  This will prevent the need for additional simplifications.

Choosing to multiply by (and not ) will create the smallest perfect square under the radical in the denominator.

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Multiplying by will create the smallest perfect cube under the radical. Replacing by 3, rationalizes the denominator.

 Make sure you multiply by whatever makes the radicand (the number under the radical sign) the smallest possible value to be simplified.  This will avoid having to further simplify later on.                 

2. When there is more than one term in the denominator, the process is a little tricky.  You will need to multiply the denominator by its conjugate.  The conjugate is the same expression as the denominator but with the opposite sign in the middle.

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Write the reciprocal of

 Be sure to enclose expressions with multiple terms in (  ). This will help you to remember to FOIL these expressions.  Always reduce the root index (numbers outside radical) to simplest form (lowest) for the final answer.  3.  When working with the reciprocal of a rational expression , if there is a radical in the denominator, you must rationalize the denominator.

An expression is in simplest radial form if:

1. The radicand of an nth root has no nth powers in it, (radical part is fully reduced)

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2. the root index is as low as possible, and  (fraction outside radical is fully reduced)3. there are no radicals in the denominator (rationalize the denominator).

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Lesson # 18 Aim: How do we complete the square?Students will be able to1. identify a perfect square trinomial2. factor a perfect square trinomial3. express a perfect square trinomial as the square of a binomial4. state the relationship between the coefficient of the middle term and the constant term of a perfect square trinomial5. determine the constant term that needs to be added to a binomial or trinomial to make it a perfect square trinomial6. solve quadratic equations by completing the square

Factoring by Completing the Square

Find the x-intercepts of y = 4x2 – 2x – 5.

First off, remember that finding the x-intercepts means setting y equal to zero and solving for the x-values, so this question is really asking you to "Solve 4x2 – 2x – 5 = 0".

This is the original problem. 4x2 – 2x – 5 = 0

Move the loose number over to the other side. 4x2 – 2x = 5

Divide through by whatever is multiplied on the square term. 

Take half of the coefficient (don't forget the sign!) of the x-term, and square it.

 Add this square to both sides of the equation.

Convert the left-hand side to squared form, and simplify the right-hand side.

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Square-root both sides, remembering the "±" on the right-hand side.  Simplify as necessary.

Solve for "x =".

Remember that the "±" means that you have two values for x.

When you complete the square, make sure that you are careful with the sign on the x-term when you multiply by one-half. If you lose that sign, you can get the wrong answer in the end. Also, don't be sloppy and wait to do the plus/minus sign until the end.

Here's another example:

Solve x2 + 6x – 7 = 0 by completing the square.

Do the same procedure as above, in exactly the same order. (Study tip: Always working these problems in exactly the same way will help you remember the steps when you're taking your tests.)

This is the original problem. x2 + 6x – 7 = 0

Move the loose number over to the other side. x2 + 6x      = 7

Take half of the x-term (that is, divide it by two) (and don't forget the sign!), and square it. Add this square to both sides of the equation.

Convert the left-hand side to squared form.  Simplify the right-hand side. (x + 3)2 = 16

Square-root both sides. Remember to do "±" on the right-hand side. x + 3 = ± 4

Solve for "x =". Remember that the "±" gives you two solutions. Simplify as necessary.

   x = – 3 ± 4      = – 3 – 4, –3 + 4      = –7, +1

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If you are not consistent with remembering to put your plus/minus in as soon as you square-root both sides, then this is an example of the type of problem where you'll get yourself in trouble. You'll write your answer as "x = –3 + 4 = 1", and have no idea how they got "x = –7", because you won't have a square root symbol "reminding" you that you "meant" to put the plus/minus in. That is, if you're sloppy, these easier problems will embarass you!

Here's another one:

Solve x2 + 6x + 10 = 0.

Apply the same procedure as before: Copyright © Elizabeth Stapel 2000-2005 All Rights Reserved

This is the original equation. x2 + 6x + 10 = 0

Move the loose number over to the other side. x2 + 6x = – 10

Take half of the coefficient on the x-term (that is, divide it by two) (don't forget the sign!), and square it. Add this square to both sides of the equation.

Convert the left-hand side to squared form.  Simplify the right-hand side.

Note: If you don't know about complex numbers yet, then you have to stop at this step, because a square can't equal a negative number! Otherwise, proceed...

(x + 3)2 = –1

Square-root both sides. Remember to put the "±" on the right-hand side.

Note the square root of a negative number!

Solve for "x =", and simplify as necessary. x = –3 ± i

If you don't yet know about complex numbers (the numbers with "i" in them), then you would say that this quadratic has "no solution". If you do know about complexes, then you would say that this quadratic has "no real solution" or that is has a "complex solution". In either case, this quadratic had no "real" solution. Since solving "(quadratic) = 0" for x is

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the same as finding the x-intercepts (assuming the solutions are real numbers), it stands to reason that this quadratic should not intersect the x-axis. This relationship is always true. If you come up with a real value on the right side (a zero value is real, by the way; the square root of zero is just zero), then the quadratic will have two x-intercepts (or only one, if you get plus/minus of zero on the right side); if you get a negative on the right side, then the quadratic will not cross the x-axis.

Lesson #19 Aim: How do we apply the quadratic formula to solve quadratic equations with rational roots?Objectives: Students will be able to: explain how the method of completing the square results in the quadratic formula, state the quadratic formula, use the quadratic formula to solve quadratic equations, express rational roots in simplest form.

Do Now: 1) Solve by factoring x2+3x-4 (x+4)(x-1)2) Solve by factoring x2-5x+6 (x-2)(x-3)3) Solve by factoring x2-10x+13=0 Cannot be factored See example 2

Homework: Page 426 # 4-6,10,11

Derive the Quadratic Formula by solving ax2 + bx + c = 0.  

This is the original equation. ax2 + bx + c = 0

Move the loose number to the other side. ax2 + bx = –c

Divide through by whatever is multiplied on the squared term.Take half of the x-term, and square it. Add the squared term to both sides.

Simplify on the right-hand side; in this case, simplify by converting to a common denominator.

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Convert the left-hand side to square form (and do a bit more simplifying on the right).

Square-root both sides, remembering to put the "±" on the right.

Solve for "x =", and simplify as necessary.

Discriminant:1) B2-4ac is a positive number that is a perfect square, the quadratic equation has 2

rational roots.2) B2-4ac =0, the quadratic equation has two equal rational roots or in effect one root.3) B2-4ac, is a positive number that is not a perfect square, the quadratic equation has

irrational roots therefore the equation can not be solved by factoring.4) B2-4ac is a negative number, the roots are imaginary.

Example 1Here are some examples of how the Quadratic Formula works: Solve x2 + 3x – 4 = 0This quadratic happens to factor: x2 + 3x – 4 = (x + 4)(x – 1) = 0.so x = –4 and x = 1. How would this look in the Quadratic Formula? Using a = 1, b = 3, and c = –4, it looks like this:

Example 2 Find the roots for 2x2+5x=12 Transform to 2x2+5x-12=0 a=2, b=5 and c=-12B2-4ac= 25-4(2)(-12)= 121 Perfect square therefore it can be solved by factoring.

Example 3Find the roots for x2-10x+13=0

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Lesson #20 Aim: How do we apply the quadratic formula to solve quadratic equations with irrational roots?

Objectives: Students will be able to state the quadratic formula. use the quadratic formula to solve quadratic equations. express irrational roots in simplest radical form. approximate irrational roots in decimal form.

When the roots of a quadratic equation are imaginary, they always occur in conjugate pairs.

 A root of an equation is a solution of that equation. 

If a quadratic equation with real-number coefficients has a negative discriminant, then the two solutions to the equation are complex conjugates of each other. (Remember that a negative number under a radical sign yields a complex number.)

The discriminant is the  b2- 4ac  part of the quadratic formula (the part under the radical sign).  If the discriminant is negative, when you solve your quadratic equation the number under the radical sign in the quadratic formula is negative --- forming complex roots.

                             Quadratic equation:  

                               Quadratic formula:         Example 1)   

Find the solution set of the given equation over the set of complex numbers. Pick out the coefficient values representing a, b, and c, and substitute into the quadratic formula, as you

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would do in the solution to any normal quadratic equation. Remember, when there is no number visible in front of the variable, the number 1 is there.

a = 1,     b = -10,     c = 34

a = 3,    b  = -4,   c  = 10

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Lesson #21 Aim: How do we apply the quadratic formula to solve verbal problems?Students will be able to1. state the quadratic formula2. create an appropriate quadratic equation that can be used to solve the verbal problem3. use the quadratic formula to identify possible solutions (roots)4. use the zero finder of a graphing calculator to verify the roots of a quadratic equation5. verify each solution in the words of the problemWriting Exercise: Tronty’s answer to his real world quadratic verbal problem gave him two roots. He rejected the negative root. Deepak

Lesson #22 Aim: What are complex numbers and their properties?Objectives: Students will be able to: define an imaginary number and a complex number; simplify powers of I; differentiate between complex and imaginary numbers; plot

points on the complex number plane; identify a complex number as a vector quantity; compute the absolute value of a number.

Do Now:

Homework: Page 924 # 10-12,17-20,36,37

The Imaginary Unit is defined as i = .It is said that the term "imaginary" was coined by René Descartes in the seventeenth century and was meant to be a derogatory reference since, obviously, such numbers did not exist.   Today, we find the imaginary unit being used in mathematics and science.  Electrical engineers use the imaginary unit (which they represent as j ) in the study of electricity. Imaginary numbers occur when a quadratic equation has no roots in the set of real numbers. A pure imaginary number can be written in bi form where b is a real number and  i   is 

Examples: pure imaginary numbers

 

  *        i =    or   - i =  - Practice Page 924 # 1-5

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Solve the following quadratic equation: x2 + 4 = 0 Remember: The Imaginary Unit is defined as i = . 

         A complex number is any number that can be written in the standard form  a  +  bi,  where a  and  b are real numbers and  i  is the imaginary unit. A complex number is a real number a, or a pure imaginary number bi, or the sum of both.  Note these examples of complex numbers written in  standard a  +  bi form:   2 + 3i,  -5 + 0i .

Complex Numbera  +  bi    a bi

7  +  2i 7 2i

1  -  5i 1 - 5i

8i 0 8i

The set of real numbers and the set of imaginary numbers are subsets of the set of complex numbers. Whenever there is a negative under the radical sign, it comes out from underneath as i. When simplifying these radical numbers in terms of i, follow the usual rules for simplifying radicals and treat the i with the rules for working with a variable.        Examples:  Number bi Form

1                6i

2                              2  x  7i                

 14i

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3             

                 

 

4                

                

                      

 

 

          

 

Lesson #23 Aim: How do we add and subtract complex numbers?Objectives: Students will be able to add and subtract complex numbers algebraically and express answers in a+bi form; add and subtract complex numbers graphically and express answers in a+bi form; find the additive inverse of complex numbers.

Do Now:

Homework:

Rule: Add like terms Find the sum of the real components. Find the sum of the imaginary components (the components with the i after them).

REMEMBER: Final answer must be in simplest  form.

Examples:

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3) Express the sum of  and in the form .

4) Add     and   .

Rules:Subtracting Rule:

*Subtract like terms* Find the difference of the real components. Find the difference of the imaginary components (the components with the i after them).REMEMBER: Final answer must be in simplest  form.

3) Subtract from .

4) Subtract from .

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1.

 

2.

 

3.

 

4.

 -1 + 6i

5.

 

6.

  

7.

 

8.

 

9.

 

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10.

 

11.

 

Add: , , and .

12.

 

Subtract from .

13.

 

From the sum and , subtract .

Lesson #24 Aim: How do we multiply complex numbers?Objectives: Students will be able to multiply and combine expressions that involve complex numbers; write the conjugate of a given complex number.

Multiplication:

Multiplying two complex numbers is accomplished in a manner similar to multiplying two binomials.  You can use the FOIL process of multiplication, the distributive property, or your personal favorite means of multiplying. 

Distributive Property:(2 + 3i) • (4 + 5i) = 2(4 + 5i) + 3i(4 + 5i)                           = 8 + 10i + 12i + 15i2

                           = 8 + 22i + 15(-1)                           = 8 + 22i -15                           = -7 + 22i    Answer Be sure to replace i2 with (-1) and proceed with  the simplification.  Answer should be in a + bi

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  form.

 The product of two complex numbers is a complex number.

(a+bi)(c+di) = a(c+di) + bi(c+di)                   = ac + adi + bci + bdi2

                      = ac + adi + bci + bd(-1)                   = ac + adi + bci - bd                   = (ac - bd) + (adi + bci)                   = (ac -bd) + (ad + bc)i    answer in a+bi formThe conjugate of a complex number a + bi is the complex number a - bi. For example, the conjugate of  4 + 2i  is  4 - 2i.   (Notice that only the sign of the bi term is changed.)

The product of a complex number and its conjugate is a real number, and is always positive.(a + bi)(a - bi) = a2 + abi - abi - b2i2

                             = a2 - b2 (-1) (the middle terms drop out) = a2 + b2   AnswerThis is a real number ( no i's ) and since both values are squared, the answer is positive.1) Multiply:   (3+5i)(3-5i) Answer (34)2) Multiply:  (8+9i)(7-3i) Answer (83+39i)3) Multiply:  (4-3i)(3-4i) Answer (-25i)

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Lesson #25 Aim: How do we divide complex numbers and simplify fractions with complex denominators?Objectives: Students will be able to write the conjugate of a given complex number; find the quotient of two complex numbers and express the result with a real denominator;express the multiplicative inverse of a complex number in standard form.

Division: When dividing two complex numbers: Write the problem in fractional form Rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator. Remember that a complex number times its conjugate will give a real number.

This process will remove the i from the denominator.) Example: 

Dividing using the conjugate:

  

 

        Answer

1) Simplify:  (2+5i)2 Answer (-21+20i)2) Simplify:  8+i(8-i) Answer (9+8i)

3) Simplify: Answer (3+2i)

4) Simplify:  Answer 35/37 + (12/37)I

5) Simplify:  Answer -5-3i

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6) Simplify:  Answer 2/15 + i/15

Lesson #26 Aim: How do we find complex roots of a quadratic equation using the quadratic formula?Objectives: Students will be able to Define the discriminant, determine the value of the discriminant, determine the nature

of the roots using the discriminant. Solve a quadratic equation that leads to complex roots, express in a + bi form.

Lesson 27: Aim: How do we use the discriminant to determine the nature of the roots of a quadratic equation?Students will be able to1. Define the discriminant2. Determine the value of the discriminant3. Determine the nature of the roots using the discriminant

Do Now:

Homework:

When the roots of a quadratic equation are imaginary, they always occur in conjugate pairs. A root of an equation is a solution of that equation. 

If a quadratic equation with real-number coefficients has a negative discriminant , then the two solutions to the equation are complex conjugates of each other. (Remember that a negative number under a radical sign yields a complex number.)The discriminant is the  b2- 4ac  part of the quadratic formula (the part under the radical sign).  If the discriminant is negative, when you solve your quadratic equation the number under the radical sign in the quadratic formula is negative --- forming complex roots.

Quadratic equation:   Quadratic formula:   

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B2-4ac>0 and a perfect square Roots of equation are real rational and unequal

B2-4ac>0 and not a perfect square Roots of equation are real, irrational and unequal

B2-4ac=0 Roots are real, rational and equal

B2-4ac<0 Roots are imaginary

Classwork:Have students determine the discriminant for each equation and tell what the roots should be. Then have them solve and prove if that is the case and what the answer is. Sketch a graph of each equation. At how many places does the graph touch the x axis?X2-2x-3=0 A=1 b=-2 c=-3 B2-4ac=(-2)2-4(1)(-3)=16 Real,rational and unequal

(3,-1)X2-6x+7=0 A=1 b=-6 c=7 B2-4ac=(-6)2-4(1)(7)=8 Real, irrational and unequalX2-4x+4=0 A=1 b=-4 c=4 B2-4ac=(-4)2-4(1)(4)=0 Real, rational and equalX2-4x+5=0 A=1 b=-4 c=5 B2-4ac=(-4)2-4(1)(5)=-4 Roots are imaginary

Example 1) Find the value of the discriminants. Describe what the roots of the answer should be. Then find the solution set of the given equation over the set of complex numbers.

Pick out the coefficient values representing a, b, and c, and substitute into the quadratic formula.HINT:  When the directions say:  Express over the set of complex numbers, look for a negative value under the radical sign.

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a = 1,     b = -10,     c = 34

a = 3,    b  = -4,   c  = 10

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Be sure to set the quadratic equation equal to 0. Arrange the terms of the equation from the highest exponent to the lowest exponent.

Lesson #28 Aim: How do we find the sum and product of the roots of a quadratic equation?Objectives: Students will be able to find the sum and product of the roots of a quadratic equation, find the missing coefficients given one root of the equation, check the solutions to quadratic equations using the sum and product relationships.

Homework:

Do Now:

What are the sum of the roots? (1) What are the product of the roots? (-20)

We can find the sum and products of the roots through the following shortcut.-b/a= 1 c/a= -20

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There is a definite relationship between the roots of a quadratic equation and the coefficient of the second term and the constant term.The sum of the roots of a quadratic equation is equal to the negation of the coefficient of

the second term divided by the leading coefficient.

The product of the roots of a quadratic equation is equal to the constant term divided by

the leading coefficient. Lets try a few of theseEquation Factors Solution Sum Product =-b/a c/aX2+5x-14 (x+7)(x-2) X=-7,2 -5 -14 -5 -14X2-4x+3 (x-3)(x-1) X=3,1 4 3 4 32x2+12x+10 (x+5)(2x+2) X=-5,-1 -6 5 -6 56x2+9x-15 (3x-3)(2x+5) X=1.-2.5 -1.5 -2.5 -9/6 -15/6X2-8x+16 (x-4)(x-4) X=4,4 8 16 8 16

Exercise: Take 2 Binomials and multiply using FOIL. Solve for the roots. Find their sum and their product. Test that –b/a and c/a will give you the same 2 answers. Do this exercise twice.

Lesson #29 Aim: How do we solve quadratic-linear systems of equations using the graphing calculator?.Students will be able to1. Explain what is meant by a system of equations and by the solution to a system of equations2. Explain how to use the graphing calculator to graph each equation3. Explain how to use the graphing calculator to solve systems of equations4. Determine the graphing window for a system of equations5. Identify the graphs from their equations6. Solve systems of equations using the graphing calculator7. Check the solution to the system of equations

Lesson #30 Aim: How do we solve quadratic-linear systems of equations algebraically? Note: This includes rational equations that can

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be written as linear equations with restricted domain, which, if not carefully considered might produce extraneous roots for the system. i.e. and y x x xy = 1 = 2 − .Students will be able to1. Explain how to solve for one variable in terms of the other2. Explain how to substitute one equation into the other to create one equation in one variable3. Algebraically solve the system of equations for all possible solutions4. Algebraically check the solutions to the system of equations5. Graphically verify the solution of a quadratic-linear system found algebraically

A quadratic equation is defined as an equation in which one or more of the terms is squared but raised to no higher power.  The general form is ax2 + bx + c = 0, where a, b and c are

constants.

In Algebra and Geometry, we learned how to solve linear - quadratic systems algebraically and graphically.  With our new found knowledge of quadratics, we are now ready to attack

problems that cannot be solved by factoring, and problems with no real solutions.

The familiar linear-quadratic system:(where the quadratic is in one variable)

Remember that linear-quadratic systems of this type can result in three graphical situations such as:

The equations will intersect in two The equations will intersect in one The equations will not intersect.

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locations.  Two real solutions. location.  One real solution. No real solutions.

Keep these images in mind as we proceed to solve these linear-quadratic systems algebraically.

Example 1:When we studied these systems in Algebra, we encountered situations that could be solved by factoring, such as this first example.

Solve this system of equations algebraically:                  y = x2 - x - 6    (quadratic equation in one variable of form y = ax2 + bx + c )                  y = 2x - 2         (linear equation of form y = mx + b)

Substitute from the linear equation into the quadratic

equation and solve.y = x2 - x - 6

2x - 2 = x2 - x - 62x = x2 - x - 4

0 = x2 - 3x - 4

0 =(x - 4)(x + 1)

x - 4 = 0      x + 1 =0x = 4           x = -1

Find the y-values by substituting each value of x into the linear  

equation.

 y = 2(4) - 2 = 6POINT (4,6)

y = 2(-1) - 2 = -4POINT (-1,-4)

See how to use your TI-83+/84+ graphing

calculator  with quadratic-linear

systems.Click calculator.

There are 2 "possible" solutions for the system:  (4,6) and (-1,-4)Check each in both equations.

y = x2 - x - 66 = (4)2 - 4 - 6 = 6 checks y = 2x - 26 = 2(4) - 2 = 6 checks

y = x2 - x - 6 -4 = (-1)2 - (-1) - 6 = -4 checksy = 2x - 2-4 = 2(-1) - 2 = -4 checks

Answer:

{(4, 6), (-1, -4)}   

Example 2:With our new found knowledge of quadratics, we are now ready to attack problems that cannot be solved by factoring, and/or problems with no real solutions (such as this second example).

Solve this system of equations algebraically:                 y = x2 - 2x + 1    (quadratic equation in one variable of form y = ax2 + bx +

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c )                 y = x - 3                (linear equation of form y = mx + b)

Substitute from the linear equation into the quadratic

equation.y = x2 - 2x + 1

x - 3 = x2 - 2x + 10 = x2 - 3x + 4

Use quadratic formula:

No real solutions.

Find the y-values by substituting each value

of x into the linear  equation.

 

POINT

POINT

There are 2 "possible" solutions. Check each in both equations.

y = x2 - 2x + 1

y = x - 3

  -------------------------------------y = x2 - 2x + 1

y = x - 3

Answer: There are no real solutions.The answers are complex numbers, which are not graphed in the Cartesian coordinate plane.

 

Other linear - quadratic systems: (where the quadratic is in two variables)

Quadratics in two variables look like x2 + y2 = 16 where two variables are squared. 

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Example 3 Solve this system of equations algebraically:                 x2 + y2 = 25  (quadratic equation of a circle center (0,0), radius 5)                 4y = 3x          (linear equation)

Substitute from the linear equation into the quadratic

equation and solve.

Find the y-values by substituting each value of x

into the linear  equation.

  

Answer:

{(4, 3), (-4, -3)}

There are 2 "possible" solutions for the system:  (4,3) and (-4,-3)Check each in both equations.

x2 + y2 = 2542 + 32 = 2516 + 9 = 2525 = 25 checks 4y = 3x4(3)=3(4)12 = 12 checks

x2 + y2 = 25(-4)2 + (-3)2 = 2516 + 9 = 2525 = 25 checks 4y = 3x4(-3)=3(-4)-12 = -12 checks 

Solve this system of equations algebraically:                 x2 + y2 = 26  (quadratic equation)                 x - y = 6          (linear equation)

Substitute from the linear equation into the quadratic

equation and solve.

Find the y-values by substituting each value of x

into the linear  equation.

Answer:

{(5, -1), (1, -5)}

There are 2 "possible" solutions for the system:  (4,3) and (-4,-3)Check each in both equations.

x2 + y2 = 2652 + (-1)2 = 2625 + 1 = 2626 = 26 checks x - y = 65 - (-1) = 66 = 6 checks

x2 + y2 = 26

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12 + (-5)2 = 261 + 25 = 2626 = 26 checks x - y = 61 - (-5) = 66 = 6 checks

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