Sine and Cosine Function Graphs - AP Precalculus

1. What is the amplitude of the function $y = -4 \, \cos(x)$?

   4. Amplitude is the absolute value of the coefficient, so $|-4| = 4$.

2. What is the vertical shift in $y = \sin(x) + 3$?

   3 units up. The constant $+3$ shifts the entire graph 3 units upward.

3. What is the range of $y = 4 \, \cos(x) + 1$?

   [-3, 5]. Range is $[1-4, 1+4] = [-3, 5]$ with amplitude 4 and vertical shift 1.

4. Identify the amplitude of $y = 10 \, \cos(x)$.

   10. Amplitude is the absolute value of the coefficient.

5. What is the standard form of the sine function?

   $y = a \, \sin(bx + c) + d$. General form where $a$ is amplitude, $b$ affects period, $c$ affects phase shift, and $d$ is vertical shift.

6. Identify the amplitude of $y = \frac{1}{2} \cos(x)$.

   $\frac{1}{2}$. Amplitude is the absolute value of the coefficient.

7. What is the period of $y = \cos(2x)$?

   $\pi$. Period equals $\frac{2\pi}{b} = \frac{2\pi}{2} = \pi$.

8. What is the period of $y = \sin(\frac{x}{2})$?

   $4\pi$. Period equals $\frac{2\pi}{b} = \frac{2\pi}{1/2} = 4\pi$.

9. What is the frequency of the function $y = \cos(4x)$?

   4. Frequency equals $\frac{b}{2\pi} \cdot 2\pi = b = 4$.

10. What is the minimum value of $y = -6 \, \cos(x)$?

    -6. For negative amplitude, minimum equals the amplitude value.

11. What is the amplitude of $y = 7 \, \sin(x)$?

    7. Amplitude is the absolute value of the coefficient.

12. What is the standard form of the cosine function?

    $y = a \, \cos(bx + c) + d$. General form where $a$ is amplitude, $b$ affects period, $c$ affects phase shift, and $d$ is vertical shift.

13. What effect does a negative coefficient have on $y = -\sin(x)$?

    Reflects over the x-axis. Negative coefficient flips the graph vertically across the x-axis.

14. Identify the phase shift in $y = \sin(x - \frac{\pi}{4})$.

    $\frac{\pi}{4}$ to the right. Phase shift is $-\frac{c}{b} = -\frac{(-\pi/4)}{1} = \frac{\pi}{4}$ to the right.

15. What is the effect of $b$ in $y = a \, \cos(bx + c) + d$?

    Affects the period. Parameter $b$ determines how many cycles occur in $2\pi$ units.

16. What is the maximum value of $y = -2 \sin(x)$?

    2. For negative amplitude, maximum equals the negative of the amplitude.

17. Identify the vertical shift in $y = \cos(x) + 4$.

    4 units up. The constant $+4$ shifts the entire graph 4 units upward.

18. What is the phase shift in $y = \sin(2x + \pi)$?

    $-\frac{\pi}{2}$. Phase shift is $-\frac{c}{b} = -\frac{\pi}{2} = -\frac{\pi}{2}$.

19. What is the effect of $d$ in $y = a \, \sin(bx + c) + d$?

    Vertical shift. Parameter $d$ moves the graph up or down vertically.

20. Identify the phase shift in $y = \cos(x + \frac{\pi}{3})$.

    $\frac{\pi}{3}$ to the left. Phase shift is $-\frac{c}{b} = -\frac{\pi/3}{1} = -\frac{\pi}{3}$ (left).

21. What is the frequency of the function $y = \sin(3x)$?

    3. Frequency equals $\frac{b}{2\pi} \cdot 2\pi = b = 3$.

22. State the phase shift formula for $y = \sin(bx + c)$.

    $-\frac{c}{b}$. Standard formula for horizontal shift in transformed sine functions.

23. State the phase shift of $y = \sin(x + \frac{\pi}{6})$.

    $\frac{\pi}{6}$ to the left. Phase shift is $-\frac{c}{b} = -\frac{\pi/6}{1} = -\frac{\pi}{6}$ (left).

24. What is the maximum value of $y = 2 \, \cos(x)$?

    2. Maximum value equals the amplitude for positive coefficient.

25. What is the frequency of $y = \sin(5x)$?

    5. Frequency equals the coefficient $b$ of $x$.

26. What is the effect of $a$ in $y = a \, \sin(bx + c) + d$?

    Affects the amplitude. Parameter $a$ determines the vertical stretch or compression.

27. What is the vertical shift in $y = \sin(x) + 6$?

    6 units up. The constant $+6$ shifts the entire graph 6 units upward.

28. State the period of $y = \cos(\frac{1}{3}x)$.

    $6\pi$. Period equals $\frac{2\pi}{b} = \frac{2\pi}{1/3} = 6\pi$.

29. Identify the period of $y = \sin(\frac{2}{3}x)$.

    $3\pi$. Period equals $\frac{2\pi}{b} = \frac{2\pi}{2/3} = 3\pi$.

30. What is the range of $y = -5 \, \sin(x) + 2$?

    [-3, 7]. Range is $[2-5, 2+5] = [-3, 7]$ with amplitude 5 and vertical shift 2.

31. What is the vertical shift in $y = \cos(x) - 2$?

    2 units down. The constant $-2$ shifts the entire graph 2 units downward.

32. What is the amplitude of $y = 3 \, \sin(x)$?

    3. The coefficient of sine gives the amplitude (distance from center to peak).

33. What is the horizontal shift in $y = \cos(x - \frac{\pi}{2})$?

    $\frac{\pi}{2}$ to the right. Phase shift is $-\frac{c}{b} = -\frac{(-\pi/2)}{1} = \frac{\pi}{2}$ to the right.

34. What is the vertical shift in $y = \sin(x) - 1$?

    1 unit down. The constant $-1$ shifts the entire graph 1 unit downward.

35. What is the period formula for a sine function $y = \sin(bx)$?

    $\frac{2\pi}{b}$. Period equals $\frac{2\pi}{b}$ where $b$ is the coefficient of $x$.

36. What is the phase shift in $y = \cos(3x - \pi)$?

    $\frac{\pi}{3}$ to the right. Phase shift is $-\frac{c}{b} = -\frac{(-\pi)}{3} = \frac{\pi}{3}$ to the right.

37. What is the range of the function $y = -3 \, \sin(x)$?

    [-3, 3]. Range is $[-|a|, |a|] = [-3, 3]$ for amplitude 3.

38. What is the range of the function $y = 2 \, \cos(x)$?

    [-2, 2]. Range is $[-|a|, |a|] = [-2, 2]$ for amplitude 2.

39. What is the maximum value of $y = 5 \, \sin(x)$?

    5. Maximum value equals the amplitude for positive coefficient.

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