4.4 Graphs of Sine and Cosine: Sinusoids

1. 4.4 Graphs of Sine and Cosine: Sinusoids

2. By the end of today, you should be able to: • Graph the sine and cosine functions • Find the amplitude, period, and frequency of a function • Model Periodic behavior with sinusoids

3. Unit Circle

4. The Sine Function: y = sin(x) • Domain: • Range: • Continuity: • Increasing/Decreasing: • Symmetry: • Boundedness: • Absolute Maximum: • Absolute Minimum: • Asymptotes: • End Behavior:

5. The Cosine Function: y = cos(x) • Domain: • Range: • Continuity: • Increasing/Decreasing: • Symmetry: • Boundedness: • Maximum: • Minimum: • Asymptotes: • End Behavior:

6. Any transformation of a sine function is a Sinusoid f(x) = a sin (bx + c) + d Any transformation of a cosine function is also a sinusoid • Horizontal stretches and shrinks affect the period and frequency • Vertical stretches and shrinks affect the amplitude • Horizontal translations bring about phase shifts

7. The amplitude of the sinusoid: f(x) = a sin (bx + c) +d or f(x) = a cos (bx+c) + d is: The amplitude is half the height of the wave.

8. y = 2 sin x y = -4 sin x You Try! y = 0.73 sin x y = -3 cos x Find the amplitude of each function and use the language of transformations to describe how the graphs are related to y = sin x

9. The period (length of one full cycle of the wave) of the sinusoid f(x) = a sin (bx + c) + d and f(x) = a cos (bx + c) + d is: When : horizontal shrink by a factor of When : horizontal stretch by a factor of If b < 0, then there is also a reflection across the y-axis

10. Find the period of each function and use the language of transformations to describe how the graphs are related to y = cos x. y = cos 3x y = -2 sin (x/3) You Try! y = cos (-7x) y = 3 cos 2x

11. The frequency (number of complete cycles the wave completes in a unit interval) of the sinusoid f(x) = a sin (bx + c) + d and f(x) = a cos (bx + c) + d is: Note: The frequency is simply the reciprocal of the period.

12. Find the amplitude, period, and frequency of the function: You Try!

13. Identify the maximum and minimum values and the zeros of the function in the interval y = 2 sin x

14. Getting one sinusoid from another by a phase shift Ex) Write the cosine function as a phase shift of the sine function Ex) Write the sine function as a phase shift of the cosine function

15. Combining a phase shift with a period change Construct a sinusoid with period and amplitude 6 that goes through (2,0)

16. Select the pair of functions that have identical graphs:

17. Select the pair of functions that have identical graphs:

18. Homework Pg. 394-395 4, 12, 16, 20, 28, 33, 37, 38, 48, 54, 56, 58, 64

19. 4.5 - Graphs of Tangent, Cotangent, Secant, and Cosecant

20. y = tan x • Domain: • Range: • Continuity: • Increasing/Decreasing: • Symmetry: • Boundedness: • Asymptotes: • End Behavior: • Period

21. Asymptotes at the zeros of cosine because if the denominator (cosine) is zero, then the function (tangent x) is not defined there. Zeros of function (tan x) are the same as the zeros of sin (x) because if the numerator (sin x) is zero, then it makes the who function (tan x) equal to zero.

22. y = cot x • Domain: • Range: • Continuity: • Increasing/Decreasing: • Symmetry: • Boundedness: • Asymptotes: • End Behavior: • Period

24. Secant Functiony = sec x • Domain: • Range: • Continuity: • Increasing/Decreasing: • Symmetry: • Boundedness: • Asymptotes: • End Behavior: • Period:

25. Cosecant Functiony = cscx • Domain: • Range: • Continuity: • Increasing/Decreasing: • Symmetry: • Boundedness: • Asymptotes: • End Behavior: • Period: