Trig/Pre-Calculus Chapter 1

1. Trig/Pre-Calculus Chapter 1

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. Finding Slope • 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.

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

5. Special Lines • Slope is __________ • Equation: _________ • Slope is __________ • Equation: _________

6. 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. 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. 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. 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?

10. 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. Function Notation Evaluating Functions = Plug AND Chug Let . Find h(1), h(-2), h(w), and h(x+1)

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

15. 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. Domain and Range

17. Increasing and DecreasingRelative Max and Min Values Increasing: Decreasing Rel Max: Rel Min: Increasing: Decreasing Rel Max: Rel Min: Increasing: Decreasing Rel Max: Rel Min:

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

19. 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. 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. 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. Common Functions Identity Function f(x)=x Abs Value Function f(x)=|x| Constant Function f(x)=c Quadratic Function f(x)=x2 Square Root Function f(x)= Cubic Function f(x)=x3

23. Vertical and Horizontal Shifts y= y= y=x2 y=x2 • Start with f(x) • Vertical Shift --> Add to or Subtract from __ • Horizontal Shift --> Add to or Subtract from __

24. Reflecting Graphs y=x+1 y=x2 y= y= • Reflection in the x-axis: h(x) = -f(x) • Reflection in the y-axis: h(x) = f(-x)

25. Nonrigid Transformations • Nonrigid - Cause a distortion

26. Nonrigid Transformations • Compare y=|x| to y=|x| y=x2 • Compare y=x2 to

27. 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. Combining Functions • Sum • Difference • Product • Quotient

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

30. 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. 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. 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. 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. 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. Verifying Inverse Functions • Inverse Functions “undo” each other, so verify that

36. 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. 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. 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