1. Panlilio 2008-2009

2. 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

3. Panlilio 2008-2009 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

4. Panlilio 2008-2009 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

5. Panlilio 2008-2009 Slope is __________ Equation: _________ Special Lines Slope is __________ Equation: _________

6. Panlilio 2008-2009 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

7. Panlilio 2008-2009 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

8. Panlilio 2008-2009 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

9. Panlilio 2008-2009 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-3-201 y47432 x-3-2-2-3 y2352

10. Panlilio 2008-2009 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”

11. Panlilio 2008-2009 Function Notation InputOutputEquation xy or f(x) Evaluating Functions = Plug AND Chug Let. Find h(1), h(-2), h(w), and h(x+1)

12. Panlilio 2008-2009 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

13. Panlilio 2008-2009 Real-Life Functions The number N (in millions) of cellular phone subscribers in the United States increased in a linear pattern from 1995 to 1997. 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 1995. Use this function to approximate the number of cellular phone subscribers for each year from 1995 to 2001.

14. Panlilio 2008-2009 Difference Quotients To Solve, Plug AND Chug! This ratio is called a difference quotient

15. Panlilio 2008-2009 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

16. Panlilio 2008-2009 Domain and Range

17. Panlilio 2008-2009 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:

18. Panlilio 2008-2009 Piecewise-Defined Functions Piecewise Function - A function that is defined by two or more equations over a specified domain

19. Panlilio 2008-2009 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

20. Panlilio 2008-2009 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.

21. Panlilio 2008-2009 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

22. Panlilio 2008-2009 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

23. Panlilio 2008-2009 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=

24. Panlilio 2008-2009 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=

25. Panlilio 2008-2009 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

26. Panlilio 2008-2009 Nonrigid Transformations Compare y=x 2 to y=x 2 y=|x| Compare y=|x| to

27. Panlilio 2008-2009 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

28. Panlilio 2008-2009 Combining Functions Sum Difference Product Quotient

29. Panlilio 2008-2009 Combining Functions For each set of equations, find (f+g)(x), (f-g)(x), (fg)(x), and (f/g)(x)

30. Panlilio 2008-2009 Composition of Functions The composition of function f with function g is: For each set of equations, find when x=0,1, and 2

31. Panlilio 2008-2009 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.

32. Panlilio 2008-2009 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

33. Panlilio 2008-2009 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:

34. Panlilio 2008-2009 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 ______

35. Panlilio 2008-2009 Verifying Inverse Functions Inverse Functions “undo” each other, so verify that

36. Panlilio 2008-2009 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

37. Panlilio 2008-2009 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

38. Panlilio 2008-2009 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