Panlilio

Section 1.1 Objectives Find the slopes of lines Write linear equations given points on lines and their slopes Use slope-intercept forms of linear equations to sketch lines Use slope to identify parallel and perpendicular lines

Panlilio Slope = Find slope for the following points: Why is Slope represented by the letter “m”? No one seems to know! One theory is that it stands for “modulus of slope”, another is that the French word for “climb” is “monster”, but nothing can be proven. Finding Slope

Panlilio There are three main “forms” for linear equations Slope-Intercept Form ___________________ Point-Slope Form___________________ Standard Form___________________ Find a linear equation given the following: Writing Equations for Lines

Panlilio Slope is __________ Equation: _________ Special Lines Slope is __________ Equation: _________

Panlilio Parallel Lines Parallel Lines have __________ slopes Write the equation for the line that passes thru (1,2) that is parallel to 4x-y=5 Write the equation for the line that passes thru (0,-4) that is parallel to -3x+4y=8

Panlilio Perpendicular Lines Perpendicular Lines have __________ slopes “Flip it and Reverse It” m --> _______ Write the equation for the line that passes thru (-4,1) that is perpendicular to -x+3y=4 Write the equation for the line that passes thru (1,5) that is perpendicular to 5x-15y=10

Panlilio Section 1.2 Objectives Decide whether relations between two variables represent a function Use function notation and evaluate functions Find the domain of functions Use the functions to model and solve real-life problems Evaluate difference quotients

Panlilio What is a function? For every ________, there is exactly one _________ Domain: Set of all _____ values Range: Set of all _____ values Does each relation represent a function? x y47432 x y2352

Panlilio Testing for Functions Algebraically Solve for y. It is a function if each x corresponds to _____ value of y. Graphically Use the “Vertical Line Test”

Panlilio Function Notation InputOutputEquation xy or f(x) Evaluating Functions = Plug AND Chug Let. Find h(1), h(-2), h(w), and h(x+1)

Panlilio Finding Domain Again, the Domain is the set of all ___ values If given a list of points, the domain is all the ________ If given an equation, find the __________ values Interval Notation: [ or ] means “includes” ( or ) means “does not include” Always use ( or ) for

Panlilio Real-Life Functions The number N (in millions) of cellular phone subscribers in the United States increased in a linear pattern from 1995 to Then, in 1998, the number of subscribers took a jump, and until 2001, increased in a different linear pattern. These two patterns can be approximated by the function Where t represents the year, with t=5 corresponding to Use this function to approximate the number of cellular phone subscribers for each year from 1995 to 2001.

Panlilio Difference Quotients To Solve, Plug AND Chug! This ratio is called a difference quotient

Panlilio Section 1.3 Objectives Find the domains and ranges of functions and use the Vertical Line Test for functions Determine intervals on which functions are increasing, decreasing, or constant Determine relative maximum and relative minimum values of functions Identify and graph piecewise-defined functions Identify even and odd functions

Panlilio Domain and Range

Panlilio Increasing and Decreasing Relative Max and Min Values Increasing: Decreasing Rel Max: Rel Min: Increasing: Decreasing Rel Max: Rel Min: Increasing: Decreasing Rel Max: Rel Min:

Panlilio Piecewise-Defined Functions Piecewise Function - A function that is defined by two or more equations over a specified domain

Panlilio Even and Odd Functions Even Odd Symmetric to _________ f(-x)=f(x) for all x’s Symmetric to _________ f(-x)=-f(x) for all x’s

Panlilio Even and Odd Functions Determine whether a function is even, odd, or neither, by evaluating f(-x). If f(-x)=-f(x), it’s ______. If f(- x)=f(x), it’s ______. If not, it’s neither.

Panlilio Section 1.4 Objectives Recognize graphs of common functions Use vertical and horizontal shifts and reflections to graph functions Use nonrigid transformations to graph functions

Panlilio Common Functions Constant Function f(x)=c Identity Function f(x)=xAbs Value Function f(x)=|x| Cubic Function f(x)=x 3 Square Root Function f(x)=Quadratic Function f(x)=x 2

Panlilio Vertical and Horizontal Shifts Start with f(x) Vertical Shift --> Add to or Subtract from __ Horizontal Shift --> Add to or Subtract from __ y=x 2 y= y=x 2 y=

Panlilio Reflecting Graphs Reflection in the x-axis: h(x) = -f(x) Reflection in the y-axis: h(x) = f(-x) y=x 2 y= y=x+1 y=

Panlilio Nonrigid Transformations Nonrigid - Cause a distortion y=cf(x)Multiply Y by Vertical ________ c > 1 Vertical ________ 0 < c < 1 y=f(cx)Multiply X by Horizontal ________ 0 < c < 1 Horizontal ________ c > 1

Panlilio Nonrigid Transformations Compare y=x 2 to y=x 2 y=|x| Compare y=|x| to

Panlilio Section 1.5 Objectives Add, subtract, multiply, and divide functions Find compositions of one function with another function Use combinations of functions to model and solve real-life problems

Panlilio Combining Functions Sum Difference Product Quotient

Panlilio Combining Functions For each set of equations, find (f+g)(x), (f-g)(x), (fg)(x), and (f/g)(x)

Panlilio Composition of Functions The composition of function f with function g is: For each set of equations, find when x=0,1, and 2

Panlilio Real-Life Compositions The number N of bacteria in a refrigerated food is given by where T is the temperature of the food in degrees Celsius. When the food is removed from refrigeration, the temperature of the food is given by Where t is the time (in hours). Find the composition N(T(t)) and interpret its meaning. Find the number of bacteria in the food when t = 2 hours. Find the time when the bacterial count reaches 2000.

Panlilio Section 1.6 Objectives Find inverse functions informally and verify that two functions are inverse functions of each other Use graphs of functions to decide whether functions have inverse functions Determine if functions are one-to-one Find inverse functions algebraically

Panlilio Finding Inverse Functions Inverse Functions: When the domain of f is equal to the ________ of f -1, and vice versa. Inverse Functions “undo” each other. Examples:

Panlilio Graphs of Inverse Functions If the point (a,b) lies on f, then the point (b,a) must lie on f -1. That means that inverse functions are symmetrical about ______

Panlilio Verifying Inverse Functions Inverse Functions “undo” each other, so verify that

Panlilio One-to-One Functions One-to-one functions: Every X has only one Y, and Every Y has only one X One-to-one functions pass the Horizontal Line Test For one-to-one functions, f(a)=f(b) implies that a=b

Panlilio Finding Inverse Functions Use the Horizontal Line Test to test whether f is a one- to-one function and has an inverse function Switch the x’s and y’s Solve for y. Replace y with f -1

Panlilio Homework 1.1: P.11 #1,19,25,33,37,43,51,53,55,65,69,83 1.2: P.24 #1,2,7,8,13,19,29,35,37,38,49,53,55,69,73, 83,86 1.3: P.38 #1,3,13-19 odd,41,45,47,49,53 1.4: P.48 #1-11 odd,15-25 odd,67,68 1.5: P.58 #5-25 EOO,35,45,47,49,51-54,57,67,69, 77,78,82 1.6: P.69 #9-13 odd,21-24,25,43,45,49,51,83 Chapter Review P.82 #1-45 EOO,47,65,69-72,85-93 odd,97,107