AP®︎ Precalculus: Topic 4.7 Practice Test

Essential Knowledge: 4.7.A.1

A parametrization $(x(t), y(t))$ for an implicitly defined function will, when $x(t)$ and $y(t)$ are substituted for $x$ and $y$, respectively, satisfy the corresponding equation for every value of $t$ in the domain.

Essential Knowledge: 4.7.A.2

If $f$ is a function of $x$, then $y = f(x)$ can be parametrized as $(x(t), y(t)) = (t, f(t))$. If $f$ is invertible, its inverse can be parametrized as $(x(t), y(t)) = (f(t), t)$ for an appropriate interval of $t$.

Essential Knowledge: 4.7.B.1

A parabola can be parametrized in the same way that any equation that can be solved for $x$ or $y$ can be parametrized. Equations that can be solved for $x$ can be parametrized as $(x(t), y(t)) = (f(t), t)$ by solving for $x$ and replacing $y$ with $t$. Equations that can be solved for $y$ can be parametrized as $(x(t), y(t)) = (t, f(t))$ by solving for $y$ and replacing $x$ with $t$.

Essential Knowledge: 4.7.B.2

An ellipse can be parametrized using the trigonometric functions $x(t) = h + a\cos t$ and $y(t) = k + b\sin t$ for $0 \le t \le 2\pi$.

Essential Knowledge: 4.7.B.3

A hyperbola can be parametrized using trigonometric functions. For a hyperbola that opens left and right, the functions are $x(t) = h + a\sec t$ and $y(t) = k + b\tan t$ for $0 \le t \le 2\pi$. For a hyperbola that opens up and down, the functions are $x(t) = h + a\tan t$ and $y(t) = k + b\sec t$ for $0 \le t \le 2\pi$.