CALCULUS CHAPTER 1 Unit Question: How do domain, range and symmetry help us to compare functions of varying degrees?

SECTION 1.1 EQ: How are equations of lines determined? EQ: What is meant by linear regression?

LINES  Increments  Δ read delta—means t he change in  Δx = x 1 - x 2  Δy = y 1 - y 2  Example find Δx and Δy Given (1, 4) (2, 5)

SLOPE POINT SLOPE EQUATIONS

USE THE POINT SLOPE TO FIND THE EQUATION  Given (-2, -1 ) and (3, 4)

SLOPE INTERCEPT FORM

EXAMPLE

GENERAL EQUATION OF A LINE

REGRESSION ANALYSIS  Given the table below, allow 1986 to be year 0. Find the linear model and use the model to predict the world population in Also, what might you have expected the population to be in YearPopulation in millions

HW PG 7 EXERCISES 4 TO 44 BY4’S + 47

SECTION 1.2 EQ: What are some common functions and how are they transformed?

FUNCTIONS AND GRAPHS DAY 1 How are functions defined and domain and range found?

FUNCTIONS AND GRAPHS DAY 1  Relation—a pairing of two items either in a set of ordered pairs or through an equation  Function—is a rule (equation) or set of ordered pairs that assigns one and only one element of the range to each element of the domain

FUNCTIONS AND GRAPHS DAY 1  Domain—the set of all 1 st components in a relation or set of ordered pairs, (x-values, independent variable)  Range—the set of all 2 nd components in a relation or set of ordered pairs, (y-values, dependent variable)

 1 st test for a function  Vertical line test  When graphed no two points fall on the same vertical line

EVALUATING THE DOMAIN AND RANGE  Natural Domain—the largest set of x-values that gives real y values and is not limited by statement or context

LIMITS ON THE DOMAIN AND RANGE

 Graphing can be a help  Looking at a table can help as well

REMEMBER FROM TRIGONOMETRY Determines width and reflection over the x-axis Left/right movement Determines up/down movement y=a function (bx-h)+k A bit different than trig: Determines width and reflection over the y-axis

TRANSFORMATIONS  Parent functions— The primary position of commonly used functions each one has at least one point that can easily be moved knowing the properties of transformations State the equation, domain and range of each as well as any asymptotes: y = 0 and x = 0

TRANSFORMATIONS State the equation, domain and range of each as well as any asymptotes:

EXAMPLES FunctionDomainRange y = x 2 ( 0,∞ ) [0, ∞ ) [-1, 1][0, 1]

 Function Notation  f(x) and y mean the same thing  evaluating a function—substituting the given value  Example  If f(x) = x find f(2)

 Composite functions  evaluating one function through another  Example  Given f(x) = x and g(x) = x + 1  Find f(g(3)) f(g(x))

HW PG 17 6 – 18 (A – C)EVENS 38, 40, 49, 50

FUNCTIONS AND GRAPHS DAY 2 How are even and odd functions determined?

EVEN AND ODD FUNCTIONS  Even or symmetric to the y-axis  If f(-x) = f(x)  Means that (x, y) maps to (-x, y)  Ex: f(x) = x 2  Odd or symmetric to the origin  if f(-x) = -f(x)  Means that (x, y) maps to (-x, -y)  Ex: f(x) = x 3

EXAMPLES—ARE THE FOLLOWING EVEN OR ODD

PIECEWISE FUNCTIONS {

EXAMPLES: GRAPH, STATE THE DOMAIN AND RANGE

{

HW PG 17 6 – 18 (D)EVENS 20 – 34 EVENS

FUNCTIONS AND GRAPHS DAY 3 How can you create a piecewise function?

FINDING PIECEWISE FUNCTIONS FROM GRAPHS  Determine the piecewise function from the graph provided

FINDING PIECEWISE FUNCTIONS FROM GRAPHS  Determine the piecewise function from the graph provided

COMPLETE THE GRAPH: a)Assuming the function is EVEN b)Assuming the function is ODD

HW PG EVENS A TO F EVENS

SECTION 1.0 EQ: How do you factor equations?

0—Make sure the terms are in order x 2, x, 1—Look for a common factor May be a number or a variable or both BASIC RULES OF FACTORING Examples: 4x 2 – 8 4(x 2 – 2) 9x 3 y 2 +15x 2 y 2 – 3xy 2 3xy 2 (3x 2 + 5x – 1)

2—if there are two terms remaining: Check to see if each is a perfect square or a perfect cube and follow the formulas Difference of two squares (must be joined by - ) a 2 – b 2 = (a + b) (a – b) Sum or Difference of two cubes a 3 ± b 3 =(a ± b)(a 2 ab+ b 2 ) BASIC RULES OF FACTORING

EXAMPLES x 8 – y 8 (x 4 + y 4 ) (x 4 - y 4 ) (x 4 + y 4 )(x 2 + y 2 )(x 2 – y 2 ) (x 4 + y 4 )(x 2 + y 2 )(x + y)(x – y) 3x y 6 3(x 6 – 8y 6 ) not perfect sqs. 3(x 2 – 2y 2 ) (x 4 + 2x 2 y 2 + 4y 4 ) x 3 – 27y 3 (x – 3y)(x 2 + 3xy + 9y 2 )

3—if there are three terms remaining: it is a trinomial make two sets of parenthesis determine which signs to place in them the factors of the first are placed in the first half of each the factors of the last are placed in the last half of each then the middle term is verified using the OI part of FOIL if the sign before the constant is - use a + and a – if the sign before the constant is + use two of the sign before the x term Or memorize Have Use

EXAMPLES x 2 + 2x – 8 (x + 4)(x – 2) 3x x ( x 2 + 5x + 6) 3 (x + 2) (x + 3) Challenge since you can’t factor out the 2 2x 2 + x – 3 (2x + 3)(x – 1)

ALTERNATE METHOD WHEN YOU CAN’T REMOVE THE LEADING COEFFICIENT  Examples: 20x 2 – 3x – 2 2 ∙ 20 = 40 factors of 40 that subtract to give you Rewrite: 20x 2 – 3x – 2 As 20x 2 + 5x – 8x – 2 Use grouping 5x(4x + 1) – 2(4x + 1) Since they share (4x + 1) Factor that out of each (4x +1)(5x - 2)

4—if there are four terms remaining: using grouping —look for terms that have a common factor (remember they may need to be rearranged)

EXAMPLES t 3 + 6t 2 – 2t – 12 t 2 (t + 6) -2 (t + 6) (t + 6) (t 2 – 2) xy + xz + wy + wz x ( y + z) + w ( y + z) ( y + z) ( x + w)

5x 3 -15x x12x x – 4 Basic rules of factoring

4x 2 – 64 3x 5 – 81x 2 90x 2 – 39x – 30

HW WORKSHEET

SECTION 1.3 EQ: What is an exponential function and how is it used to solve real-world problems?

EXPONENTIAL FUNCTIONS

EXAMPLES

EXPONENTIAL FUNCTIONS

GROWTH AND DECAY  Exponential growth happens when a>1  Exponential decay happens when 0 < a < 1 Graph y = 3 x and y = 5 x

EXTENSION  Given y = 2 x y = 3 x and y = 5 x  Determine for what values of x is: 2 x < 3 x < 5 x for what values of x is: 2 x > 3 x > 5 x for what values of x is: 2 x = 3 x = 5 x

EXPONENTIAL REGRESSIONS Is the following data linear? And how can you tell? Exponential Regressions Formula f(x) = P a x P == the initial amount a == 1 + rate of growth as a decimal x ==# of time units 1)Find the exponential regression 2)What will the population be in a)2010b) )Check your phone what was the population in )AP Stats folks—is this close enough, why might it be off? What is the difference called? YearWorld Population in millions Verify the slope between points Remember 1986 = 0 7 billion or million y = (1.019)

EXPONENTIAL DECAY  Suppose the half life of a certain radioactive substance is 20 days and that there are 5 grams present initially. When will there only be 1 gram remaining? f(x) = P a x f(t) = 5 (1-.5) t f(t) = 5 (.5) t f 2 (t) = 1 Intersect when t = time units is 2.32(20) = 46.4 days

FIND THE GROWTH RATE

HW PG EVEN AND 24, 26, 28, 34