Pre Calc P3,4 Bl 4.notebook

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January 30, 2013

Jan 19­12:31 PM

Welcome to classRm 410Ms. Dougherty

When you come into class take two books:PreCalculus and Functions Modeling ChangeComplete the information sheet

Jan 19­12:42 PM

Goal of PreCalculus: Graphical, Numerical, Algebraic• To help students to truly understand the

fundamental concepts of algebra, trigonometry, and analytic geometry.

• To foreshadow important ideas of calculus• To show how Algebra and Trigonometry

can be used to model real­life problems.

Jan 19­12:40 PM

What is a function?Graphical:

Numerical:

Algebraic

Jan 19­12:53 PM

A Function represents the relationship between two quantities in which every element in the domain has a unique element in the range. For every input, there is only one output.

When is a relationship not a function?Give a numerical, graphical and an algebraic example. (3 separate and different examples)

Group Work:

Jan 19­1:02 PM

Classwork; FMC p. 7 #1,4,9,10. p31 #8

Homework/CW PreCalc p74 #1-10

Jan 19­1:04 PM

p 6 1. low temperature on July 19thb. When 73 degrees?c. Is the daily low temperature a function of the date? Why/why not??d. Is the date a function of daily low temperature ? Why/why not??

Pre Calc P3,4 Bl 4.notebook

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January 30, 2013

Jan 19­1:09 PM

4. a 9. a 10. ab bi b.c bii c.

c d.e.

p 7

p 31 #8 situation table graph

a.b.c.

d.

Jan 19­1:17 PM

HomeWork p74 #1-10

Table equation graph

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Jan 19­1:27 PM

Day 2 ­ Warm UpBob purchases a house 8 years ago for 42,000. This year it was appraised for 67,000.a. Write a linear model of the situation.

b. Graph the equation and trace to estimate in how many years will the house be worth 72,500.

c. Write and solve an equation algebraically to determine how long it will take before the house is worth 74,000.

d. worth 80,250.

Jan 19­7:59 PM

Goal: Linear ReviewEquations:

a. Slope ­intercept formy = mx + b; m= (y2­y1)/(x2­x1)

b = y­interceptUse this form when you know slope and y­intercept

b. Point slope formy2­y1 =m(x2­x1)Use this form when you know slope and a point. Will be the most useful form when doing Calculus

c. Standard FormAx + By = CUse this form when you need a quick graph. Set x=0 to get the y­intercept and set y = 0 and find x­intercept

d. Special Equationsy = b; Horizontal linex = a; Vertical line

Jan 19­8:16 PM

Parallel Lines: two lines are parallel if m1 = m2

Decide which of the following are parallel 3x + 2y =6

y = 2x + 6y - 3 = -2/3(x - 6)10 -2y = 3xy - 1 = - 2 (x-3)

Perpendicular Lines: two lines are perpendicular if m1 = -1/m2

Jan 19­8:24 PM

Classwork FMC: p 50 #1,4,7,10,13,14,21,22,27*

Homework p 59 # 3,7,8,11,13,14

* Use the parabola y = x2 ­ 1 instead of x2 + 1 in # 27

Pre Calc P3,4 Bl 4.notebook

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January 30, 2013

Jan 24­1:40 PM Jan 24­1:45 PM

Jan 24­1:57 PM Jan 24­11:52 AM

Factor the followingx2-16 x2 +10x +25 81x2 - 4 3x3-15x2+ 18x

16h4 - 81 x2 + 3x - 4 2x2-11x +5

x2 +2xh + h2 x2 -3x +4 x4 + x2 -20

Jan 19­8:28 PM

Day 2: Polynomial ReviewWarm Up: Consider a ball being thrown up from a bridge and allowed to fall pass the bridge all the way to the ground t seconds after it is released.Let h(t) = -16t2 + 56t + 120 How high is the ball when released? When does the ball hit the ground? Sketch the graph and state the domain and range.

Jan 25­1:14 PM

Pre Calc P3,4 Bl 4.notebook

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January 30, 2013

Jan 25­1:21 PM Jan 19­8:31 PM

Quadratic 1. Equations

a. Standard Equationy = Ax2 + Bx + C

b. Vertex Formy - k = A(x- h)2

vertex (h, k)2. Graph: parabola

A > 0, opens upwardA < 0, opens downward

3. Symmetrysymmetric about y=h

Jan 19­8:44 PM

Solving equations using graphs, tables and algebraic manipulationZero Property of MultiplicationIf ab = 0 then a = 0 or b = 0

Exploration 1 p 41 (P5)4x2 - 12x + 7 = 0Do 1-7

Jan 19­9:42 PM

Solving Quadratic EquationsThere are four basic ways to solve quadratic equationsalgebraically1. Factoring Example 2x2 + 3x - 2 = 0 (2x + 1)(x - 2) = 0 2x + 1 = 0 or x - 2 = 0 x = -1/2 or x = 2

Jan 19­9:27 PM

2. Solving by Extracting Square Roots(2x - 1)2 = 92x - 1 = + 32x = 4 or 2x = -2x = 2 x = -1

Jan 22­2:44 PM

2. Completing the SquareTo complete the square

Divide through by AAdd (B/2)2 and subtract (B/2)2

Ex. 4x2 - 20x + 17 = 0 x2 - 5x + 17/4 = 0

x2 - 5x + (-5/2)2 = -17/4 +(-5/2)2

(x - 5/2)2 = 2 x - 5/2 = +√2 x = 5/2 +√2

Pre Calc P3,4 Bl 4.notebook

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January 30, 2013

Jan 19­9:46 PM

3. Quadratic Equation x = -b + √(b2 - 4ac) 2a

ex 3x2-6x =5a = 3, b = -6, c = -5x = 6 + √(-6)2 - 4(3)(-5) or x = 6 - √(-6)2 - 4(3)(-5) 2(3) 2(3)x = 6 + √96 =2.63 or x = 6 + √96 = -0.63

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Jan 19­9:55 PM

Classwork/ HomeworkPreCalc p 45 # 4,5,8,11,18,19-24,31,

37, 45, 47-54,68

ClassWork #68

Jan 22­2:14 PM

Warm Up Interpretation of functions

Jan 22­2:50 PM

A Function is a rule that takes every value in the domain and assigns to each value exactly one output. The output (also known as the Range) if a function of the input (Domain)

Goal: Describe a function

Jan 22­3:01 PM

Find the domain of the following functions:f(x) = √x+3g(x) = (√x) / (x - 5)h(x) = √3 /4 s2 , where s is the side length of an equilateral triangle

Domain: input, x-valuesImplied domain: defines the functionRelevant domain: fits the situation

Jan 29­1:06 PM

Pre Calc P3,4 Bl 4.notebook

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January 30, 2013

Jan 22­3:06 PM

List ways in which you could describe a graph

Jan 22­3:07 PM

Domain and RangeRange: all the y coordinates that correspond with the points in the domain.

Find the range of the functionsf(x) = √x+3g(x) = (√x) / (x - 5)h(x) = √3 /4 s2 , where s is the side length of an equilateral triangle

Precalc p 96 #1-8, State the domain

Jan 29­1:16 PM Jan 29­1:42 PM

Jan 22­3:10 PM

Contiuity:A function is contiuous at a point c if the function does not come apart at that point.What are some ways that you have seen a discontinuity?

P 96 #13

Jan 22­3:11 PM

A Function is increasing or decreasing? What do we call it when it is neither?

Increasing or decreasing throughout?

p 96 #14

Pre Calc P3,4 Bl 4.notebook

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January 30, 2013

Jan 22­3:16 PM

Local and Absolute ExtremaA local Maximum of a function f is a value f(c) that is greater than or equal to all the range values of f on some interval containing c. (If it is > all range values then f(c) is the absolute maximum

A local Minimum of a function f is a value f(c) that is less than or equal to all the range values of f on some interval containing c. (If it is < all range values then f(c) is the absolute minimum

Jan 22­3:14 PM

p 96 9-12,15-18, 19, 20, 29, 30, 33,34

Jan 24­2:23 PM

Goal : Piecewise FunctionsEven odd functionsTen Basic Graphs

Jan 24­2:24 PM

29. f(X) = l x+2 l -1

In order to work with absolute value equation we need to rewrite the expression without absolute value bars

y = x + 2 -1 when x > -2ory = -( x+2) -1 when x < 2

x+1, when x > -2

-x -3, when x < -2

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THIS IS CALLED A PIECEWISE FUNCTIONS BECAUSE..........

Another look at Problem #29 &30

Jan 24­2:41 PM

When different formulas are used on different parts of the domain of f, the function is said to be PIECEWISE DEFINEDAbsolute Value equations are piecewise defined

Take Problem # 30 from last nights home work and rewrite f(x) as a piecewise function

Classwork FMC p 93 # 4,5

Jan 22­3:24 PM

Symmetry -looks the same With respect to the y-axis

Graphically Numerically Algebraic

Functions with this property are even functions.

Pre Calc P3,4 Bl 4.notebook

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January 30, 2013

Jan 22­3:27 PM

Symmetry -looks the same With respect to the x-axis

Graphically Numerically Algebraic

Jan 22­3:27 PM

Symmetry -looks the same With respect to the origin

Graphically Numerically Algebraic

Functions with this property are odd functions.

Jan 22­3:30 PM

pg 97 #35-42p105 #1-20

Pre-Calc

Jan 24­3:11 PM

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f(x) =x f(x) =x2

f(x) =x3 f(x) = 1/x

Jan 24­2:22 PM

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f(x) =abs(x) f(x) =√x

f(x) =ex f(x) = ln x

Jan 24­3:36 PM

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f(x) =sinx

f(x) =cos x