Sinusoidal Functions - AP Precalculus

1. Maximum occurs when $\sin(x) = 1$: $8 + 5(1) = 13$.

Average value of max and min. The horizontal line around which the function oscillates.

1. The coefficient of cosine gives the amplitude.

Vertical shift. $D$ moves the entire graph up or down from the x-axis.

$C$. The horizontal shift is $C$ units to the right when positive.

-4. The constant term gives the vertical shift downward.

$\frac{\pi}{4}$. Phase shift is the value subtracted inside the function.

-5. Minimum occurs when sine equals -1: $2(-1) - 3 = -5$.

1. The coefficient of cosine gives the amplitude.

[-8, -4]. Range is $[-6-2, -6+2] = [-8, -4]$.

$8\pi$. Period = $\frac{2\pi}{\frac{1}{4}} = 8\pi$.

1. Amplitude is the absolute value: $|-4| = 4$.

Amplitude. $A$ controls the maximum distance from the sinusoidal axis.

$y = A , \text{sin}(B(x - C)) + D$. Standard form with amplitude $A$, frequency $B$, phase shift $C$, and vertical shift $D$.

$[-8, -4]$. Range is $[-6-2, -6+2] = [-8, -4]$

$\frac{2\pi}{3}$. Period = $\frac{2\pi}{3}$ when $B = 3$.

[-1, 5]. Range is $[2-3, 2+3] = [-1, 5]$ since amplitude is 3.

$\frac{\pi}{2}$. Phase shift is $\frac{\pi}{2}$ units to the right.

$y = -1$. The sinusoidal axis is at $y = D = -1$.

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

1. The coefficient 7 in front of cosine is the amplitude.

-4. The constant term gives the vertical shift downward.

$y = 5 , \text{sin}(2x)$. Amplitude 5 and period $\pi$ means $B = \frac{2\pi}{\pi} = 2$.

$-\pi$. Rewrite as $\sin(x - (-\pi))$, so phase shift is $-\pi$.

1. Amplitude is the absolute value of the coefficient: $|-2| = 2$.

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

-5. Minimum occurs when sine equals -1: $2(-1) - 3 = -5$.

1. Maximum occurs when cosine equals 1: $3(1) + 1 = 4$.

[-6, 2]. Range is $[D-|A|, D+|A|] = [-2-4, -2+4]$.