AP®︎ Precalculus: Topic 3.6 Practice Test

Essential Knowledge: 3.6.A.1

Functions that can be written in the form $f(\theta) = a\sin\bigl(b(\theta + c)\bigr) + d$ or $g(\theta) = a\cos\bigl(b(\theta + c)\bigr) + d$, where $a$, $b$, $c$, and $d$ are real numbers, $a \ne 0$, and $b \ne 0$, are sinusoidal functions and are transformations of the sine and cosine functions. Additive and multiplicative transformations are the same for both sine and cosine because the cosine function is a phase shift of the sine function by $-\frac{\pi}{2}$ units.

Essential Knowledge: 3.6.A.2

The graph of the additive transformation $g(\theta) = \sin\theta + d$ of the sine function $f(\theta) = \sin\theta$ is a vertical translation of the graph of $f$, including its midline, by $d$ units. The same transformation of the cosine function yields the same result.

Essential Knowledge: 3.6.A.3

The graph of the additive transformation $g(\theta) = \sin(\theta + c)$ of the sine function $f(\theta) = \sin\theta$ is a horizontal translation, or phase shift, of the graph of $f$ by $-c$ units. The same transformation of the cosine function yields the same result.

Essential Knowledge: 3.6.A.4

The graph of the multiplicative transformation $g(\theta) = a\sin\theta$ of the sine function $f(\theta) = \sin\theta$ is a vertical dilation of the graph of $f$ and differs in amplitude by a factor of $|a|$. The same transformation of the cosine function yields the same result.

Essential Knowledge: 3.6.A.5

The graph of the multiplicative transformation $g(\theta) = \sin(b\theta)$ of the sine function $f(\theta) = \sin\theta$ is a horizontal dilation of the graph of $f$ and differs in period by a factor of $\left|\frac{1}{b}\right|$. The same transformation of the cosine function yields the same result.

Essential Knowledge: 3.6.A.6

The graph of $y = f(\theta) = a\sin\bigl(b(\theta + c)\bigr) + d$ has an amplitude of $|a|$ units, a period of $\left|\frac{1}{b}\right|2\pi$ units, a midline vertical shift of $d$ units from $y = 0$, and a phase shift of $-c$ units. The same transformations of the cosine function yield the same results.