Praxis Prep

1. Praxis Prep Graphs of Trig Functions By Tim Nov. 15, 2007

2. Hard Problem

3. Sine Curve • Dom of sine function is all real numbers. Range is [-1, 1] • Period is 2pi. Sine curve is symmetric w.r.t the origin. • X-intercepts are at 0, pi, and 2pi

4. Cosine Curve • Dom of cosine function is all real number. Range is [-1, 1] • Period is 2pi. Cosine curve is symmetric w.r.t the y-axis. • X-intercepts are at pi/2 and 3pi/2

5. Demonstration – Unit Circle and Trig curves Go to the website listed below Select Sin and Start http://www.intmath.com/Trigonometric-graphs/1_Graphs-sine-cosine-amplitude.php Record values of t as the sin wave crosses the x-axis and when the wave reaches its max and min values.

6. Notice From Demonstration That the shapes of the sine and cosine curves are regular (they repeat after the wheel has gone around once)? We say such curves are periodic. The period is the time it takes to go through one cycle and then start over again. That the sine and cosine graphs are almost identical, except shifted sideways from each other? That in the interactive, the radius of the circle is 80 units and the curve went up to 80 units and down to -80 units on the vertical axis? This quantity of a sine and cosine curve is called the amplitude of the graph. This indicates how much energy the graph has. Higher amplitude means greater energy. http://www.intmath.com/Trigonometric-graphs/1_Graphs-sine-cosine-amplitude.php

7. PRS Praxis • The best representation to highlight periodic behavior is a • Line graph • Paragraph describing the amplitude • Trigonometric equation • Diagram of the unit circle

8. PRS Praxis • The best representation to highlight periodic behavior is a • Line graph • Paragraph describing the amplitude • Trigonometric equation • Diagram of the unit circle • Things that occur regularly are good candidates for modeling with trig equations. Many biological rhythms can be modeled with sine and cosine function.

9. Amplitude The a in both of the graph types y =a sinx andy =a cosx affects the amplitude of the graph. The amplitude is the distance from the "resting" position (otherwise known as the mean value or average value) of the curve. In the interactive simulation, the amplitude was 80 units. In the unit circle the amplitude is 1. Amplitude is always a positive quantity. We could write this using absolute value signs. For the curves y = a sin x and y = a cos x, amplitude = |a|

10. Graph of Sine x - with varying amplitudes We start with y = sin x. It has amplitude = 1 and period = 2π. Now let's look at y = 5 sin x. I have used a different scale on the y-axis. This time we have amplitude = 5 and period = 2π.

11. And now for y = 10 sin x. Now, amplitude = 10 and period = 2π. For comparison, and using the same y-axis scale, here are the graphs of p(x) = sin x, q(x) = 5 sin x and r(x) = 10 sin x on the one set of axes. Note that the graphs have the same period (which is 2π) but different amplitude.

12. Graph of Cosine x - with varying amplitudes Now let's have a look at the graph of y = cos x. We note that the amplitude = 1 and period = 2π.

13. Similar to what we did with y = sin x above, we now see the graphs of p(x) = cos x q(x) = 5 cos x r(x) = 10 cos x on one set of axes, for comparison:

14. PRS Question Consider: y = 3 cos 8x What is the amplitude of this function ? A. 1/3 B. 30 C. D. 3

15. PRS Question • Consider: y = 3 cos 8x • What is the amplitude of this function ? • 1/3 B. 30 C. 2pi / 8 D. 3 • In the periodic function y = a cosbx • The amplitude or “height” or “strength” of the curve is denoted by a.

16. Graphs of y = a sin bx and y = acosbx • The b in both of the graph types • y = a sin bx • y = a cos bx • affects the period (or wavelength) of the graph. The period is the distance (or time) that it takes for the sine or cosine curve to begin repeating again. • The period is given by: • Note: As b gets larger, the period decreases.

17. Changing the Period First, let's look at the graph of y = 10 cos x, As we learned, the period is 2π

18. Now let's look at y = 10 cos (3x). Note the 3 inside the cosine term. Notice that the period is different. (The amplitude is 10 in each example.) This time the curve starts to repeat itself at x = 2π/3. This is consistent with the formula for period:

19. Now let's view the 2 curves on the same set of axes. Note that both graphs have an amplitude of 10 units, but their period is different. Note:b tells us the number of cycles in each 2π. For y = 10 cos x, there is one cycle between 0 and 2π (because b = 1). For y = 10 cos 3x, there are 3 cycles between 0 and 2π (because b = 3).

20. In General If 0 < b < 1, the period of y = sin bx is greater than 2pi and represents a horizontal stretching of the graph. If b >1, period of y = cosbx is less than 2pi and represents a horizontal shrinking of the graph.

21. Ex: y = sin 5x, =  Horizontal Shrinking Ex: y = cos 10 x =  Horizontal Shrinking Ex: y = cos =  Horizontal Stretching

22. PRS Question Consider: y = 3 cos 8x What is the period of this function ? A. B. C. D.

23. PRS Question Consider: y = 3 cos 8x What is the period of this function ? A. B. C. Period = 2pi/b Here b = 8. How many cycles between [0-2pi] D.

24. TI Time Graph 2 cycles of y = 3 cos 8x  Here, b = 8, so the period is 2π/8 = π/4. To draw 2 cycles, we will need to graph from 0 to π/2 along the x-axis.

25. Ti Time Note that there are 8 cycles between 0 and 2π.

26. Ti Time Graph 2 cycles of y = 4 sin x/3  Keep in mind what we’re doing…Horizontal____ Remember that the period is the distance it takes the sine or cosine curve to begin repeating again.

27. Graphs of y = a sin(bx + c) and y = acos(bx + c) Here we meet the following 2 graph types: y = a sin(bx + c) y = a cos(bx + c) Both b and c in these graphs affect the phase shift (or displacement), given by:

28. The phase shift is the amount that the curve is moved in a horizontal direction from its normal position. The displacement will be to the left if the phase shift is negative, and to the right if the phase shift is positive. There is nothing magic about this formula. We are just solving the expression in brackets for zero; bx + c = 0.

29. Ti Time Graph the curve y = sin(2x + 1) Let’s begin by graphing y = sin 2x Period is π so we can capture two cycles using Solve 2x + 1 = 0 and x = -1/2 Hence phase shift to the left by -1/2 Now graph y = sin(2x +1)

30. Ti Time Now let's consider the phase shift. Using the formula above, we will need to shift our curve by: This means we have to shift the curve to the left (because the phase shift is negative) by 0.5. Here is the answer (in blue). I have kept the original y = sin 2x (in dotted gray) so you can see what's happening.

31. Vertical Translation The final type of translation is the vertical translation caused by the constant d in the equations: y = d + a sin (bx - c) y = d + a cos (bx - c) Graph y = 3 cos 2x Now graph y = 2 + 3 cos 2x What happens?

32. Hard Problem

33. Practice 1. Computer the amplitude of Hint: We’re not looking for amplitude at a point. Amplitude refers to the strength of the curve at it’s maximum value. When do sin x and cos x have their maximum value? 2. Consider the periodic function Compute the period: Compute the amplitude: Compute the phase shift: How many cycles in 12π: How many cycles in 2π: