Functions & Graphs

1. Functions & Graphs • What is a function? • Describes a relationship of input and output • g(x) = x5 • Function name: g • Input: x • Output: input (x) raised to the fifth power • The function g takes an input (x) and raises it to the fifth power. • p(q) = 2q – 5 • The function p takes an input (q) and multiples it by 2, subtracts 5. 4.5 Graphs of Sine and Cosine

2. g(t) = sin(t) • the function g takes an input of t and outputs the sine ratio that corresponds with t (y/r) • h(t) = cos(t) • the function h takes an input of t and outputs the cosine ratio that corresponds with t (x/r) 4.5 Graphs of Sine and Cosine

3. "Why didn't sin and tan go to the party?" "... just cos!" 4.5 Graphs of Sine and Cosine

4. Think Graphically • What is the graphical representation of a function? • g(x) = 3x5+4 • A plot of all of the points (input, output) (x, g(x)) • g is the function that raises x to the 5th power and v. stretches c=3 and v. shift up 4 • p(q) = 2q – 5 • A plot of all the points (q, p(q)) • p is the function that takes q and v. stretches c=2 and vertically shifts down 5. 4.5 Graphs of Sine and Cosine

5. Generalize • f(x) = x • g(x) = a f(b(x – h)) + k • g(x) = a(b(x – h)) + k • This is similar to point-slope form: • y – y1 = m(x - x1) 4.5 Graphs of Sine and Cosine

6. g(x) = a(bx - h) + k • How does a affect the f(x)? • Vertical stretch/compression. • a > 1 v. stretch or 0 < a < 1 v. compression • How does b affect the f(x)? • Horizontal stretch/compression. • 0 < b < 1 h. stretch or b > 1 h. compression • How does h affect the f(x)? • Horizontal shift. • How does k affect the f(x)? • Vertical shift. 4.5 Graphs of Sine and Cosine

7. Predict Graph and Verify • Y1 = x2 • Y2 = (1/4x)2 • Y3 = (4x)2 • Y4 = (1/4)(x)2 • Y5 = 4(x)2 • Y6 = (x - 4)2 • Y7 = (x + 4)2 • Y8 = x2 - 4 • Y9 = x2 + 4 4.5 Graphs of Sine and Cosine

8. Ferris Wheel • Which trig function could describe the height of the Ferris Wheel at time t? • What does the height of the ferris wheel correspond to with respect to ordered pairs (x, y)? • f(t) = sin(t) • What are some key values of sine or key points of the graph of sine? • 0, max, 0, min (periodic) 4.5 Graphs of Sine and Cosine

9. Given a function g(t) = sin(t) • h(t) = d + ag(b(x – c)) • h(t) = d + asin(b(x – c)) • How does a, b,c and d affect the graph? • Remember what a graph is: • The plot of all of the points (input, output) • (t, g(t)) 4.5 Graphs of Sine and Cosine

10. Pre-calculus4.5 Graphs of Sine and Cosine Objectives: Sketch the graphs of basic sine and cosine functions Use amplitude and period to sketch the graphs Sketch translations of graphs

11. You Try Complete each t-chart and sketch the graphs: 4.5 Graphs of Sine and Cosine

12. Basic Sine and Cosine Curves What symmetry do you notice with the above graphs? How does graph symmetry relate to the ideas of even and odd functions? 4.5 Graphs of Sine and Cosine

13. Example • Sketch the graph of y = 2sin(x) on the interval [-π, 4π] • How is this similar to the parent function sin(x)? • How is this different to the parent function sin(x)? 4.5 Graphs of Sine and Cosine

14. Amplitude and Period • The amplitude is half the distance between the maximum and minimum values of the function • How does amplitude relate to listening to music? • The period is the distance in x needed for the function to complete one cycle (when the values begin to repeat). • How would changing the period affect the music? • Pitch, speed, tempo. 4.5 Graphs of Sine and Cosine

15. Standard Form of Sine and Cosine Functions • The amplitude = |a| • The period = 2π/b • Horizontal shifts are caused by c • Interval left endpoint: solve bx – c = 0 • Interval right endpoint: solve bx – c = 2π • Vertical shifts are caused by d • Max value = d + |a|, min value = d - |a| 4.5 Graphs of Sine and Cosine

16. Steps to Graph • Find left endpoint • bx – c = 0 • Find right endpoint • bx – c = 2π • Divide interval into four equal parts • Interval length is the period 2π/b • Apply basic sine shape (0, max, 0, min, 0) or basic cosine shape (max, 0, min, 0, max) with amplitude and vertical shift to get key points 4.5 Graphs of Sine and Cosine

17. Example • Graph • Endpoints: • Period: • Amplitude: • Key points: 4.5 Graphs of Sine and Cosine

18. Example • Graph • Endpoints: • Period: • Amplitude: • Key points: 4.5 Graphs of Sine and Cosine

19. Example • Graph • Endpoints: • Period: • Amplitude: • Key points: 4.5 Graphs of Sine and Cosine

20. Example • Find the amplitude, period, and phase shift for the sine function whose graph is shown. Write an equation for this graph. • Amplitude: • Period: • Phase shift: 4.5 Graphs of Sine and Cosine

21. Closure • Describe the basic shape of the sine graph and the cosine graph • Sine starts at zero going up • 0, max, 0, min, 0 • Cosine starts at max going down • max, 0, min, 0, max 4.5 Graphs of Sine and Cosine

22. 13 60° Bellwork What are the 6 trig functions for this angle? 4.5 Graphs of Sine and Cosine

23. You Try • Quickly evaluate each of the following: • One half of one half = • One half of one third= • One half of two thirds= • One half of five sixths= • One half of one half of three= • One half of one half of seven= • One half of one half = • One half of one third= • One half of two thirds= • One half of five sixths= • One half of one half of three= • One half of one half of seven= 4.5 Graphs of Sine and Cosine

24. Solutions • One half of one half = 1/4 • One half of one third= 1/6 • One half of two thirds= 1/3 • One half of five sixths= 5/12 • One half of one half of three= 3/4 or .75 • One half of one half of seven= 7/4 or 1.75 • Quickly evaluate each of the following: • One half of one half = • One half of one third= • One half of two thirds= • One half of five sixths= • One half of one half of three= • One half of one half of seven= 4.5 Graphs of Sine and Cosine