Review of Quadratic Formula

The quadratic formula is derived from completing the square on the general equation: 2 0ax bx c

You MUST memorize the formula: 2 4

2

b b acx

a

Process: 1. Write the equation in standard form: 2 0ax bx c 2. Identify , , and a b c . 3. Substitute numbers into formula. 4. Carefully do the arithmetic under the square root sign. 5. If possible, simplify the radical. 6. If possible, reduce the fraction.

1. 23 2 4 0x x 2. 28 5( 1)x x 3. 4 ( 1) 5 0x x

Quadratic Types of Equations

ReviewofFactoring:In previous math classes, you have learned to solve quadratic equations by the factoring

method. 24 8 3 0x x 25 19 4 0x x

QuadraticTypesofEquations:We have equations that look like a quadratic, but have different exponents. Some

examples of these equations are:

4 24 8 3 0x x 2 1

3 35 19 4 0x x 2 16 7 3 0x x Solve by factoring:

1. 4 24 8 3 0x x 2. 2 1

3 35 19 4 0x x

3. 2 16 7 3 0x x 4. 1 1

2 44 9 2 0x x

Equations with Fractional Exponents

ReviewofExponents:Remember that a fractional exponent can be written in radical form.

2

3 32x x

52 2

5x x

If you encounter an equation that has a variable raised to a fractional exponent, you solve it by raising both sides to the appropriate power.

2

3 32x x

52 2

5x x

Solve:

5. 3

2 26 7 27x x 6. 2

32 9x

Try these on your own:

43 16x

321 125x

Functions

A function is a set of ordered pairs where for every x-value there is a unique y-value. The x-values are called the domain (left to right). The y-values are called the range (bottom to top). The vertical line test can determine if a graph is a function or not. If a vertical line only crosses the graph once, then the graph is a function.

Determine whether the following is a function. Give the domain and range of each relation.

{(10,8), (6, 4), (2,0), ( 2, 4)}I {(3, 4), (3, 2), (8,9), (1,0)}K

Properties of Graphs 1. Specific Values 2. Domain and Range 3. Intercepts-where the graph crosses the axes. Find the following: Use the graph to find:

( 2)f (3)f

For what value(s) of x does ( ) 4f x ? For what value(s) of x does ( ) 0f x ? Find the x-intercept(s):_____________ Find the y-intercept:_____________

Determine the domain, range, any intercepts and values.

( 2)f =? ( 1)f =? (2)f =?

Given ( ) 2 5f x x , evaluate each function at the given values and simplify answers.

( 3)f (2)f ( 1)f x

Given 2( ) 3 1f x x x , evaluate each function at the given values and simplify answers. ( 3)f ( )f x ( 5)f x

Given 2

1

xh x

x

, evaluate each function at the given values and simplify answers.

1h 4h h x

Properties of Functions

Increasing/DecreasingIntervals: The part of the DOMAIN where y-values are increasing/ decreasing.

Relative Maxima: A point where a function changes from increasing to decreasing is called a relative maximum. Relative Minima: A point where a function changes from decreasing to increasing is called a relative minimum.

With the given graph, 1. Determine the domain: 2. Determine the range:

3. Determine ( 8)f

4. Solve ( ) 10f x

5. Find the intervals where the function is increasing:_____________ 6. Find the intervals where the function is decreasing:_______________ 7. Find the intervals where the function is constant:_______________ 8. Find the numbers at which f has a relative maximum:_____________ 9. Find the relative maxima:_____________ 10. Find the numbers at which f has a relative minimum:_____________ 11. Find the relative minima:_____________ 12. Find all intercepts:______________ 13. Find the values of x for which ( ) 0f x 14. Find the zeros of f

With the given graph, 1. Determine the domain:_________ 2. Determine the range:____________ 3. Find the intervals where the function is increasing: _____________ 4. Find the intervals where the function is decreasing: _______________ 5. Find the intervals where the function is constant: _______________

6. Find the numbers at which f has a relative maximum:_____________ 7. Find the relative maxima:_____________ 8. Find all intercepts:____________ 9. Find the values of x for which ( ) 0f x 10. Find the zeros of f

EvenandOddFunctionsandSymmetryAn even function is symmetric with respect to the y-axis.

Algebraically, the function f is an even function if ( ) ( )f x f x for all x in the domain of f.

An odd function is symmetric with respect to the origin.

Algebraically, the function f is an odd function if ( ) ( )f x f x for all x in the domain of f.

Determine whether each of the following functions is even, odd, or neither:

3( ) 6f x x x 4 2( ) 2f x x x 2( ) 2 1f x x x

Piecewise Functions

A piecewise function is a function in which the formula used depends upon the domain the input lies in. We notate this idea like:

formula 1 if domain to use formula 1

( ) formula 2 if domain to use formula 2

formula 3 if domain to use formula 3

f x

A cell phone company uses the function below to determine the cost, C, in dollars for g gigabytes of data transfer.

25 0 2( )

25 10( 2) 2

if gC g

g if g

Find the cost of using 1.5 gigabytes of data, and the cost of using 4 gigabytes

of data. Evaluate each piecewise function at the given values.

2

1 0( )

0

x if xf x

x if x

a) ( 2)f b) (2)f c) (0)f

0( )

1 0

x if xf x

if x

a) ( 2)f b) (2)f c) (0)f

The Difference Quotient

The difference quotient is defined by:

( ) ( )

, 0f x h f x

hh

Find the difference quotient of the following functions: 1. ( ) 5 1f x x 2. 2( ) 2 6f x x x 3. 2( ) 2 3 5f x x x

FindingSlope

Find the slopes of the lines passing through the following points.

Slope is defined as the ratio of a change in y to a corresponding change in x. For a linear function, slope may be interpreted as the rate of change of the dependent variable per unit change in the independent variable.

Formula(s) for slope: 2 1

2 1

y ym

x x

or

ym

x

or rise

mrun

Find the slopes of the lines passing through the following points. Ex #1: (7,0) and (0,4) Ex #2: ( 2, 5) and (1,9) Ex #3: (3, 5) and ( 1, 5) Ex #4: (7, 2) and (7,5)

Equations of Lines

Find the equation of a line given the slope and a point. Find the equation of the line with the given information. Write answers in slope-intercept form, if possible. You will need to know 2 formulas:

1. Slope-intercept formula: y mx b

2. Point-Slope formula. 1 1( )y y m x x

Ex. #1: 5m ; through ( 2,1) Ex. #2: 3

5m ; through ( 4, 2)

Horizontal Vertical

Equation: y number 0m only has a y-intercept

Equation: x number m is undefined only has an x-intercept

Ex. #3: 0m ; through ( 5,3)

Ex. #4: m is undefined; through ( 2, 7)

Find the equation of the line passing through the given points.

1. Find the slope first. 2 1

2 1

y ym

x x

2. Pick one point and now use the Point-Slope formula. 1 1( )y y m x x 3. Write answers in slope-intercept form, if possible.

Ex. #5: Passing through the points ( 1,3) and (4,7)

Ex. #6: Passing through the points (3, 4) and ( 5, 1)

Ex. #7: 2 and 1x intercept y intercept

Ex. #8: Passing through the points (5, 6) and ( 3, 6)

Ex. #9: Passing through the points ( 7, 4) and ( 7,8)

AverageRateofChange

For a non-linear function, the average rate of change between any two points is the slope of the line containing the two points. This line is called a secant line.

2( ) 2 3f x x x 2 1

2 1

( ) ( )y f x f x

x x x

Find the average rate of change from (a) 1 to 1 (b) 2 to 5

Find an equation of the secant line containing ( 1, ( 1))f and (3, (3))f

3 21 1( ) 2

2 2f x x x x

Find an equation of the secant line containing ( 1, ( 1))f and (2, (2))f 22 2

( )3 3

f x x x

Graphing Lines

Graphing lines using the slope and y-intercept. 1. Solve the equation for y. 2. Identify m and b. 3. Plot b on the y-axis.

4. From b, use the slope rise

run

to get more points.

Ex. #1: 4 1x y Ex. #2: 2 3 9x y

Ex. #3: 7

22

y x Ex. #4: 2y

Ex. #5: 3 12 0x

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Graph: 22 1 3f x x . Find the axis of symmetry, the domain, and the range.

Graph: 212 4

2f x x . Find the axis of symmetry, the domain, and the range.

Intercepts: 

Where the graph intersects each axis.

How to find:

Look at the graph—does not always give exact answer Let 0x and solve for y Let 0y and solve for x.

Find the intercepts:

212 4

2f x x

Find the intercepts:

211 5

3f x x

Find the intercepts:

22 1 3f x x

QuadraticFunctionsinGeneralForm

General Form: Standard Form:

Need to find the vertex: ( , )h k

Use the formula:

Graph: 2 6 7f x x x . Find the axis of symmetry, the domain, and the range.

Find the intercepts.

2( )f x ax bx c 2( ) ( )f x a x h k

, ( )2

bh k f h

a

Graph: 2 4 2f x x x . Find the axis of symmetry, the domain, and the range.

Find the intercepts.

Graph: 22 4 1f x x x . Find the axis of symmetry, the domain, and the range.

Find the intercepts.

Quadratic Inequalities

The standard form of a quadratic equation is: 2 0ax bx c A quadratic inequality replaces the equal sign with inequality signs: , , ,

Process: 1. Write the given inequality in standard form. 2. Solve the corresponding equation by factoring or the quadratic formula. 3. Plot the answers on a number line. 4. Use test points or a graph to determine what interval solves the inequality. 5. Write answers in interval notation.

1. 2 3 10 0x x 2. 22 3 5 0x x

3. 26 14 0x x 4. 2 2 1 0x x 5. 2 2 1 0x x

6. 2 3 5 0x x **

AbsoluteValueFunctions

Absolute Function in General/Standard Form:

______________________________

Vertex (or turning around point) = ____________

The slope of right branch is ________________

The slope of left branch is ________________

When we graphed quadratic functions, we followed these steps:

Find the vertex. Determine the value of “a”. Plot the vertex. From the vertex, go over one unit (to the right and left) and then up or down “a”.

Applying this technique, graph the following functions. State the vertex, the domain, and the range.

4f x x

2f x x

1 3f x x

2 1f x x

3 2 1f x x

3 5f x x

Intercepts: 

Where the graph intersects each axis.

How to find:

Look at the graph—does not always give exact answer Let 0x and solve for y Let 0y and solve for x.

Find the intercepts:

1 3f x x

Find the intercepts:

3 2 1f x x

Find the intercepts:

2 1 4f x x

Absolute Value Inequalities

Review:1. Graphing horizontal lines. 2. Graphing absolute value functions. 3. Solving quadratic inequalities.

2 1 2f x x

4, 3f x f x

2 3 10 0x x

SolvingAbsoluteValueInequalitiesProcedure:1. Sketch a graph—one for each side of the inequality. 2. Solve the corresponding equation to see where the two graphs intersect. 3. Plot answers on a number line 4. Shade the appropriate region(s). 5. Write answer in interval notation.

1. 4x

2. 2 1x

3. 2 3 5 11x

Hint: It might be easier if the absolute value expression is isolated on one side.

4. 3 1 2 9x

5. 5 13 7 6x 6. 3 1 5 9x

Graphs of Polynomial Functions

In order to sketch a graph of a polynomial function, we need to look at the “end behavior” of the graph and the intercepts. The “end behavior” of the graph is determined by the leading term of the polynomial.

4 2( ) 4 3 2 5f x x x x

3 2( ) 5 7 2 5f x x x x

2 2( ) 3(5 1) ( 2)f x x x

x

y

x

y

x

y

2( ) ( 4) ( 2)f x x x

Summary: If the leading coefficient has an even power, then the end behavior is the same: both up or both down If the leading coefficient has an odd power, then the end behavior will be opposites: one up and one down

Intercepts:In order to find the y-intercept, set x = 0 and solve for y. In order to find the x-intercept, set y = 0 and solve for x. 2( ) ( 2) ( 1)f x x x x 3 2( ) 3 3f x x x x

x

y

“Multiplicity” of the x-intercepts 1. Multiplicity of 1 or single: the graph passes through the x-axis like a line. 2. Multiplicity of 2 or even: the graph passes “bounces” off the x-axis like a parabola. 3. Multiplicity of 3 or odd: the graph “squiggles” through the x-axis like a cubic function.

Graphthefollowing: End Behavior, y-intercept, the multiplicity of the x-intercepts.

2 3( ) ( 2) ( 1)( 3)f x x x x

4 3( ) 3 3f x x x

x

y

x

y

4 2( ) 4 3 1f x x x

2 2( ) (3 1) ( 1)f x x x

3( ) 4 12f x x x

x

y

x

y

x

y

Synthetic Division and Remainder Theorem

Synthetic Division is a condensed method of long division. It is quick and easy. Unfortunately, it can only be used when the divisor is in the form of ( )x a Synthetic division:

1.

29 5 1

1

x x

x

2.

3 125

5

x

x

3.

35 2 4

1

x x

x

RemainderTheorem: When dividing a polynomial by ( )x c , then the remainder is ( )f c .

If 3( ) 5 2 4f x x x , find (1)f . (refer to #3) If 3 2( ) 11 7 19f x x x x , find ( 1)f .

Reminders: 1. Write both polynomials in standard form. 2. Fill in all missing terms with a place holder of zero. 3. Write your answer as a polynomial that is one degree less than the dividend (numerator).

If 4 3 2( ) 12 6 5f x x x x , find 2

3f

.

FactorTheorem: Let ( )f x be a polynomial. Then if ( ) 0f c , then ( )x c is a factor of ( )f x . And if ( )x c is a factor of ( )f x , then ( ) 0f c . (refer to #2) Solve the equation 3 22 3 11 6 0x x x , given that 3 is a zero (or factor) of the function

3 2( ) 2 3 11 6f x x x x Solve the equation 4 3 23 17 19 21 18 0x x x x , given that 3 is a zero of multiplicity of two of the function 4 3 2( ) 3 17 19 21 18f x x x x x

Zeros of Polynomial Functions

Some polynomials cannot be factored by traditional methods. The Rational Zero or Root Theorem gives us another method to find the x-intercepts or zeros of a polynomial. The theorem states that a list of possible rational zeros of a polynomial can be found by taking the factors of the constant term (p) and dividing them by the factors of the leading coefficient (q).

Make a list of possible rational zeros: 1. 3 2( ) 4 15f x x x 2. 5( ) 10 2 5f x x x

After making the list, we can use it along with synthetic division and the graph of the function to try and factor the polynomials and find the roots (zeros).

3 2( ) 3 5 4 4f x x x x

4 3 2( ) 6 11 21 3f x x x x x

4 3 2( ) 2 7 5 13 3f x x x x x

4 3 2( ) 3 3 16 12f x x x x x

5 3 2( ) 8 8 1f x x x x

Domain of a Function

Finding the domain of a function: 1. The implied domain is the set of all real numbers for which the expression is defined. For all polynomial functions the domain is all real numbers or expressed in interval notation: , Ex. #1: 2( ) 5 6f x x x

But what about rational functions? The question one must ask when finding the domain is “Where is this function NOT defined?” 2. Rational Functions: This is a function that is comprised of a ratio of 2 polynomial functions. Thus, there is a denominator involved. We must always remember that

DIVISION BY ZERO IS UNDEFINED!! To determine the domain of a rational function, set the denominator equal to zero and solve. These are the values that are NOT acceptable.

Ex. #2: 2

1( )

3 40f x

x x Ex. #3:

2 2

1 1( )

9 9f x

x x

Ex. #4:

1( )

53

2

f x

x

3. Radical Functions: This is a function that is underneath some type of radical. Remember when taking the EVEN root of a negative number, the answer is imaginary. Imaginary numbers are NOT acceptable for real valued functions. To determine the domain of a radical function, set the radicand greater than or equal to zero and solve. These are the values that ARE acceptable. Ex. #5: ( ) 2f x x Ex. #6: ( ) 8 6f x x

Ex. #7: ( ) 3 5f x x x

Ex. #8:

3( )

6x

f xx

Ex. #9:

2 5( )

4x

f xx

Graphing Rational Functions

A rational function is the ratio of two polynomial functions. In order to sketch a graph, we must find all the intercepts and all the asymptotes.

Process for sketching a graph of a rational function: 1. Find the y-intercept by setting x = 0. 2. Find the x-intercept(s) by setting y = 0. 3. Find the vertical asymptotes by setting the denominator = 0. (The denominator of the reduced function.) 4. Find the horizontal asymptote (if one exists) by comparing degree of the numerator to the degree of

the denominator. 5. Plot the intercepts and graph the asymptotes. Plot a few additional points to complete the graph.

1. 3

( )2

xf x

x

y-intercept: x-intercept(s): Vertical Asymptote(s): Horizontal Asymptote: If the degree of the numerator is equal to the degree of the denominator, then the horizontal asymptote will be:

coefficient of the leading term of the numerator

coefficient of the leading term of the denominatory

2. 2

2 3( )

6

xf x

x x

y-intercept: x-intercept(s): Vertical Asymptote(s): Horizontal Asymptote: If the degree of the numerator is less than the degree of the denominator, then the horizontal asymptote will ALWAYS be __________________

3. 2

2

4 1( )

2

xf x

x x

y-intercept: x-intercept(s): Vertical Asymptote(s): Horizontal Asymptote:

Polynomial Inequalities

Process: 1. Write the given inequality in standard form: exponents in descending order and zero on the right hand side. 2. Solve the corresponding equation by factoring. 3. Plot the answers on a number line. 4. Use test points OR a graph to determine what interval solves the inequality. 5. Write answers in interval notation.

1. ( 4)( 1)( 7) 0x x x 2. 2( 3)( 4)( 1) 0x x x

3. 2 2( 2) ( 4)( 6) 0x x x 4. 3 218 30 8 0x x x 5. 3 24 4 0x x x 6. 3 25 9 45 0x x x

Rational Inequalities

Process: 1. Make one side of the inequality zero. 2. Combine all of the terms on the non-zero side into a single fraction. 3. Set both the numerator and denominator EQUAL to zero and solve these equations. These are the boundary points. 4. Plot these points on a number line. 5. Look at corresponding graph and shade either above or below x-axis. 6. Write answers in interval notation.

1. 3

02

x

x

2. 2 3

14

x

x

3. 2 1

1 2

x

x

4. 2 1

5 3x x

Graph Transformations

Transformations: ( ) ( ) constantg x f x Moves the graph __________________________ ( ) ( ) constantg x f x Moves the graph __________________________ ( ) ( constant)g x f x Moves the graph __________________________ ( ) ( constant)g x f x Moves the graph __________________________ ( ) ( )g x f x Multiplies all the y-values by ___________ ( ) (Constant) ( )g x f x Multiplies all the y-values by ________________ ( ) a ( b) cg x f x a and c affects the y-values. b affects the x-values Use the graph of ( )y f x to obtain the following graphs: 1. ( ) ( ) 2g x f x 2. ( ) ( 2)g x f x

3. ( ) ( )g x f x 4. ( ) ( 2) 2g x f x

5. 1

( ) ( )2

g x f x 6. ( ) 2 ( )g x f x

7. 1

( ) ( 1) 22

g x f x

8. ( ) 1 3f x x 9. 3( ) 2( 1) 1f x x

10. 31( ) 3 2

2f x x

Basic Functions Linear

1. y x 2. 2y x 3. 2y x

Quadratic

4. 2y x 5. 22y x 6. 22y x

Absolute Value

7. y x 8. 2y x 9. 1

2y x

Square Root 10. y x 11. 2y x 12. 2y x

13. y x 14. 2y x Cubic

15. 3y x 16. 3 2y x 17. 32 2y x

Cube Root

18. 3y x 19. 3y x 20. 31

2y x

Piecewise Functions

GraphingPiecewiseFunctionsTo graph a piecewise defined function, choose several values for each domain including the endpoints of each domain, whether or not that the endpoint is included in the domain.

1. 2 1 0

( )3 0

x if xf x

if x

2. 2 1

( )1 1

x if xf x

x if x

3. 3 2 2

( )3 2

x if xf x

if x

4.

3 4 0

( ) 2 0

0

x if x

f x if x

x if x

5. 3

4 4

( ) 4 1

1

x if x

f x x if x

x if x

Composite Functions

( ) 3 1f x x 2( ) 2g x x

x ( )f x

2

5

6

TheCompositionofFunctions The composition of the function f with g is denoted by f g and is defined by the

equation: ( ) ( ( ))f g x f g x . The domain of the composite function f g is the set of

all x such that x is in the domain of g and ( )g x is in the domain of f. For 1 & 2, use functions above: 1. (2)f g 2. (2)g f

For 3-9, use graph: 3. ( )( 3)f g

4. ( )( 5)g f

5. ( )(0)f g

6. ( )(3)g f

7. ( )(7)f g

8. ( )(2)f f

**9. ( )( 7)g f

x ( )g x

2

5

6

Find ( )f g x and ( )g f x

9. Given ( ) 3 1f x x and 2( ) 2g x x

10. Given 2( )f x x and ( ) 2g x x

Find ( )f g x :

11. Given ( )3

xf x

x

and

2( )g x

x

y

x

y

x

y

x

y

x

y

x

y

x

Inverse Functions

Review:DefinitionofaFunctionA function is a set of ordered pairs where for every x-value there is a unique y-value. Graphically: Use the vertical line test to determine if the graph is a function. Determine if the following graphs are functions?

OnetoOneFunctions(1–1functions)A function is said to be one to one if each y-value corresponds to only one x-value. Graphically: Use the horizontal line test to determine if the following functions are One to One Functions. Determine if the following funcitons are One to One Functions

WhatisanInverseFunction? Only one-to-one functions have inverse functions. A function and its inverse can be described as the "DO" and the "UNDO" functions. A function takes a starting value, performs some operation on this value, and creates an output answer. The inverse of this function takes the output answer, performs some operation on it, and arrives back at the original function's starting value.

( ) 2 3f x x The inverse of ( ) 2 3f x x ? Domain of f Range of f

DefinitionofInverseFunctions:

If f is a one to one function, then 1( )f x is the inverse function of f if: 1( ( ))f f x x , for every x in the domain

And 1( ( ))f f x x , for every x in the domain

VerifyingInverses:Find ( ( ))f g x and ( ( ))g f x to determine whether each pair of functions f and g are inverses of each other.

1. 9

( )4

xf x

and ( ) 4 9g x x

2. 3

( )2

xf x

and ( ) 3 2g x x

Note: 1f x is notation for the inverse of f .

Pronounced: “ f inverse of x ”.

1f x is NOT “ f to the negative 1 exponent”

3. 2

( )5

f xx

and 2

( ) 5g xx

FindingtheInverseFunction Process: 1. Replace ( )f x with y . 2. Switch x and y . 3. Solve for y .

4. Re-write y as 1( )f x . For the following problems:

a) Find the inverse of the given function. b) VERIFY your equation is the inverse by showing 1( ( ))f f x x and 1( ( ))f f x x .

1. ( ) 3 1f x x 2. 3( ) 4f x x

3. ( ) 5f x x 4. 4

( ) 9f xx

5. 2 3

( )5

xf x

x

GraphingInverseFunctionsIn order to graph the inverse of a function, you need to switch the domain and the range. In other words, reverse the order of the ordered pairs.

ExponentialFunctions

An exponential function is a function where a positive number is raised to a power.

( ) xf x b where b > 0 and b ≠1

Exponential Functions NOT Exponential Functions

Reviewofexponents:

0x 1x 1x 1

a

b

2x

GraphthefollowingExponentialFunctions:

( ) 2xf x 1

( )2

x

f x

Domain:___________________ Domain:___________________

Range:____________________ Range:____________________

X-int:_____________________ X-int:_____________________

Y-Int:_____________________ Y-Int:_____________________

Asymptote:________________ Asymptote:________________

Transformations

( ) xf x b

All exponential functions (in “basic” form) have 2 point on their graphs:

(1, ____) (0, ____)

Vertical Shift ( ) xf x b c

( ) xf x b c

Horizontal Shift ( )( ) x cf x b ( )( ) x cf x b

Reflections ( ) xf x b

( ) xf x b

Vertical Stretch and Compress ( ) xf x c b

Note that ( )x xc b cb

TransformationsExamples:

Sketch ( ) 3xf x Then, sketch the following

4( ) 3xf x

( ) 3 4xf x

( ) 3xf x

( ) 2 3xf x

NaturalNumbere:Of all possible choices of bases, the most preferred or most natural base it the number e. The number e has important significance in science and mathematics. It is often called Euler’s number named after Leonhard Euler.

2.7182818284590452353602874713527...e

The number e is defined by:

If n is a positive integer, then 1

1n

en

as n (discussed in Calculus)

NOTE: The number e is a number, not a variable.

Sketch ( ) xf x e Then, sketch the following

4( ) xf x e

( ) 3xf x e

( ) 3 xf x e

1( ) 2xf x e

GraphsofLogarithmicFunctions

Review: Find the inverse of 2

( )3

xf x

Find the inverse of ( ) 2xf x

The exponential function has an inverse called _______________________

DefinitionofLogarithmicFunction:

For 0x and 0, 1b b , yb x is equivalent to logby x

The function ( ) logbf x x is the logarithmic function with base b.

GraphthefollowingInverseFunctions:

( ) 2xf x 2( ) logf x x

Domain:___________________ Domain:___________________

Range:____________________ Range:____________________

X-int:_____________________ X-int:_____________________

Y-Int:_____________________ Y-Int:_____________________

Asymptote:________________ Asymptote:________________

GraphthefollowingInverseFunctions:

( ) 4xf x 4( ) logf x x

Domain:___________________ Domain:___________________

Range:____________________ Range:____________________

x-int:_____________________ x-int:_____________________

y-int:_____________________ y-int:_____________________

Asymptote:________________ Asymptote:________________

Note: Some bases are used frequently, and have simplified notation. Common Log (Base 10): 10log x = Natural Log (Base e): loge x =

Inverse Functions Logarithm Form Exponential Form ( ) xf x b

( ) 10xf x

( ) xf x e

Transformations:

Parent Function: logbf x x All logarithmic functions (in “basic” form) have 2 points on their

graphs: (1,0) and ( ,1)b

Vertical Shift logbf x x c

logbf x x c

Horizontal Shift logbf x x c

logbf x x c

Reflections logbf x x

Vertical Stretch and Compress

logbf x c x

Sketch ( ) logf x x Then, sketch the following

( ) log 2f x x

( ) log( 2)f x x

( ) log( 1) 4f x x

( ) 2logf x x

( ) log( 2) 3f x x

Sketch ( ) lnf x x Then, sketch the following

( ) ln 3f x x

( ) ln 3f x x

( ) ln( 2) 4f x x

1( ) ln2f x x

( ) ln( 1) 2f x x

Properties of Logarithms

Review of exponent properties:

32 05 7

3 11 1

6

Compare exponents to logarithms:

Exponential Form Log Form

32 8

2 13

9

81 9

3 27 3

2log 16 4

3

1 1log

23

Evaluate the following expressions:

7log 49 3log 27 6

1log

6 3

1log

9

6log 6 3

1log

3 81log 9 12log 1

Common logarithms: Base Ten

2log10

5log 10

Natural Logarithms: Base e

2lne

1ln

e ln e

Since logs and exponents are inverse functions, they undo one another.

The following properties show this:

log xa a x and

loga xa x

25log 5 3ln e 7log 57

ln8e 4log 114 log1000

PropertiesofLogarithms For 0M and 0N :

1. Product Rule: log log logb b bM N MN

2. Quotient Rule: log log logb b b

MM N

N

3. Power Rule: log log pb bp M M

Use the properties of logs to expand each expression as much as possible.

3

4

2loga

p

qr

2

53

loga

u

v

Express as a single logarithm:

33log 2log log

2a a ax y z 2log log ( 3) log ( 3)a a ax x x

Exponential Equations

An exponential equation has the variable in the exponent. The easiest way to solve

is to use the same base. ( x ya a )

Process: 1. Isolate the base. 2. Re-write each side of the equation with the same base. 3. Equate the exponents. 4. Solve.

Solve the following:

2 16x 2 14 64x

1 316 8x x 1

497

x

3 23

1xee

2 19 27x

An exponential equation has the variable in the exponent. When you cannot get the bases to be the same, you have to use logarithms to solve. We make use of the following logarithmic property:

log log pa ap M M or ln ln pp M M

Process:

1. Isolate the base. 2. Take the log (ln) of both sides of the equation. 3. Use the log property (above) to re-write the exponents as coefficients. 4. Solve. 5. Use a calculator to approximate the solution.

Solve the following:

7 60x 2 140 8 x

2 1 7 13xe 35 11 35x

1 54 3x x 1 2 53 7x x

Logarithmic Equations—Form 1

log log constantb bM N 1. Combine logs using log properties. 2. If the coefficients do not equal to one, then use the log property:

log log pb bp M M

2. Now your equation should have the form of: log constantb D

3. Re-write the equation in exponential form to get rid of the log: constantb D 4. Solve for variable. 5. Check your answer back into the original equation. (You can only take the log of numbers that are greater than zero.)

Solve the following:

5log (3 2) 2x 4 2 3ln( x )

log( 2) log( 1) 1x x 2

2 2log ( 3 ) log ( 2) 2x x x

Logarithmic Equations—Form 2

log log logb b bM N C 1. Combine logs using log properties. 2. If the coefficients do not equal to one, then use the log property:

log log pb bp M M

2. Now your equation should have the form of: log logb bD C

3. Using property of equality, you can now say that D C 4. Solve for variable. 5. Check your answer back into the original equation. (You can only take the log of numbers that are greater than zero.)

2 2 2log log ( 5) log 6x x ( 4) ( 1)ln x ln x ln x 2log log7 log100x 3 3 3log (4 ) log ( 8) log (2 13)x x x

( 4) ( 1) (3 12)ln x ln x ln x

Matrices

Review:

Solve the system by elimination method:   2 9

4 3 14

x y

x y

 

 

 

 

We will be using matrices to solve Systems of Linear Equations.   A matrix is a rectangular array of numbers arranged in rows and columns placed inside brackets. The numbers inside the brackets of a matrix are called elements. 

        Example:

13 15 21

7 5 6

12 1 41

 

 An Augmented Matrix is a matrix that is used to represent a system of Linear Equations. It has a vertical bar separating the columns of the matrix into 2 groups (one for the coefficients of the variables in the linear system and the other for the answers).    o Note: If a variable is missing, we assign it the coefficient of ZERO.   Rewrite the following system of equations in an augmented matrix System  

System of Equations  Augmented Matrix 

3 9

3 3

2 0

x y z

x y z

x y z

 

      

3 5 12

4 5

2 3 4

x z

y z

x y

 

      

One method to solve a system of linear equations using a matrix is to get the matrix in Echelon Form, which means to have only 1s in your main diagonal (going from upper‐left to lower‐right) and 0s below the ones.  

Once you are in Echelon Form, you will use back substitution to solve your system.  

    Example:  

1 2 1 0

0 1 2 3

0 0 1 2

 

 

RowOperations

There are 3 row operations that produce matrices that represent systems with the same solution set. 

Row Operation   Notation   Example  

Interchange 2 Rows  1 3R R   1 1 3 3

3 18 12 21

1 2 1 0

 Multiply a Row by a non‐zero number  

2 22R R   1 2 1 0

3 18 12 21

1 1 3 3

 Multiply a Row by a non‐zero number and then add the product to any other row  

2 3 316 R R R   1 2 1 0

6 36 24 42

1 1 3 3

 

 To solve a system of linear equations using matrices we would  

1. Write system as an augmented matrix  2. Use row operations to get row equivalent matrix in Row Echelon Form  3. Use Substitution to solve for the variables. Answer solution in an ordered triple.  

Solve the following System of Equations: 

1. 3 9

3 3

2 0

x y z

x y z

x y z

2. 2 2

2 2 6

3 2 15

x y z

x y z

x y z

 

Midpoint and Distance

Distance Formula: Use to find the distance between two points .

Example 1: Find the distance between A(4,8) and B(1,12)

Example 2: Find the distance between A ( 1,4) and B (3, 2)

2 22 1 2 1distance ( ) ( )x x y y

Midpoint Formula: Use to find the center of a line segment

Example 3: Find the midpoint of A ( 2, 1) and B ( 8,6)

2 1 2 1midpoint ,2 2

x x y y

Circles

Circles

Definition of Circle: A circle is a set of points in a plane that are located a fixed

distance, called the radius from a given point in the plane called the center.

Standard Form of the Equation of a Circle:

The standard form of the equation of a circle with

the center ( , )h k and the radius r is

2 2 2( ) ( )x h y k r

Example 1: Write the equation of the circle in standard form given:

Center (2, 1) and 4r

Example 2: Write the equation of the circle in standard form given:

Center ( 5,3) and 3 2r

Example 3: Give the center and radius of the circle:

2 2( 1) ( 4) 25x y

Example 4: Give the center and radius of the circle:

2 2( 3) 20x y

Example 5: Give the center and radius of the circle:

2 2 2 8 6 0x y x y

Example 6: Give the center and radius of the circle:

2 2 8 2 19 0x y x y

Example 7: Find an equation, in standard form, of the circle that satisfies the

given conditions.

a) Center (4, 2) and tangent to the

x-axis.

b) Center ( 3,5) and tangent to the

y-axis.

c) Center at the origin; passes through (4,1)

d) Center is (6,7) ; passes through (1, 3)

e) Endpoints of the diameter are ( 8,1) and (2,7)

f) Endpoints of the diameter are ( 9, 8) and ( 3,0)