AP®︎ Precalculus: Topic 3.8 Practice Test

Essential Knowledge: 3.8.A.1

Given an angle of measure $\theta$ in standard position and a unit circle centered at the origin, there is a point, $P$, where the terminal ray intersects the circle. The tangent function, $f(\theta)=\tan\theta$, gives the slope of the terminal ray.

Essential Knowledge: 3.8.A.2

Because the slope of the terminal ray is the ratio of the change in the $y$-values to the change in the $x$-values between any two points on the ray, the tangent function is also the ratio of the sine function to the cosine function. Therefore, $\tan\theta=\frac{\sin\theta}{\cos\theta}$, where $\cos\theta\ne 0$.

Essential Knowledge: 3.8.B.1

Because the slope values of the terminal ray repeat every one-half revolution of the circle, the tangent function has a period of $\pi$.

Essential Knowledge: 3.8.B.2

The tangent function demonstrates periodic asymptotic behavior at input values $\theta=\frac{\pi}{2}+k\pi$, for integer values of $k$, because $\cos\theta=0$ at those values.

Essential Knowledge: 3.8.B.3

The tangent function increases and its graph changes from concave down to concave up between consecutive asymptotes.

Essential Knowledge: 3.8.C.1

The graph of the additive transformation $g(\theta)=\tan\theta+d$ of the tangent function $f(\theta)=\tan\theta$ is a vertical translation of the graph of $f$ and the line containing its points of inflection by $d$ units.

Essential Knowledge: 3.8.C.2

The graph of the additive transformation $g(\theta)=\tan(\theta+c)$ of the tangent function $f(\theta)=\tan\theta$ is a horizontal translation, or phase shift, of the graph of $f$ by $-c$ units.

Essential Knowledge: 3.8.C.3

The graph of the multiplicative transformation $g(\theta)=a\tan\theta$ of the tangent function $f(\theta)=\tan\theta$ is a vertical dilation of the graph of $f$ by a factor of $|a|$. If $a<0$, the transformation involves a reflection over the $x$-axis.

Essential Knowledge: 3.8.C.4

The graph of the multiplicative transformation $g(\theta)=\tan(b\theta)$ of the tangent function $f(\theta)=\tan\theta$ is a horizontal dilation of the graph of $f$ and differs in period by a factor of $\left|\frac{1}{b}\right|$. If $b<0$, the transformation involves a reflection over the $y$-axis.

Essential Knowledge: 3.8.C.5

The graph of $y=f(\theta)=a\tan(b(\theta+c))+d$ is a vertical dilation of the graph of $y=\tan\theta$ by a factor of $|a|$, has a period of $\left|\frac{1}{b}\right|\pi$ units, is a vertical shift of the line containing the points of inflection of the graph of $y=\tan\theta$ by $d$ units, and is a phase shift of $-c$ units.