AP®︎ Precalculus: Topic 1.7 Practice Test

Essential Knowledge: 1.7.A.1

A rational function is analytically represented as a quotient of two polynomial functions and gives a measure of the relative size of the polynomial function in the numerator compared to the polynomial function in the denominator for each value in the rational function’s domain.

Essential Knowledge: 1.7.A.2

The end behavior of a rational function will be affected most by the polynomial with the greater degree, as its values will dominate the values of the rational function for input values of large magnitude. For input values of large magnitude, a polynomial is dominated by its leading term. Therefore, the end behavior of a rational function can be understood by examining the corresponding quotient of the leading terms.

Essential Knowledge: 1.7.A.3

If the polynomial in the numerator dominates the polynomial in the denominator for input values of large magnitude, then the quotient of the leading terms is a nonconstant polynomial, and the original rational function has the end behavior of that polynomial. If that polynomial is linear, then the graph of the rational function has a slant asymptote parallel to the graph of the line.

Essential Knowledge: 1.7.A.4

If neither polynomial in a rational function dominates the other for input values of large magnitude, then the quotient of the leading terms is a constant, and that constant indicates the location of a horizontal asymptote of the graph of the original rational function.

Essential Knowledge: 1.7.A.5

If the polynomial in the denominator dominates the polynomial in the numerator for input values of large magnitude, then the quotient of the leading terms is a rational function with a constant in the numerator and nonconstant polynomial in the denominator, and the graph of the original rational function has a horizontal asymptote at $y = 0$.

Essential Knowledge: 1.7.A.6

When the graph of a rational function $r$ has a horizontal asymptote at $y = b$, where $b$ is a constant, the output values of the rational function get arbitrarily close to $b$ and stay arbitrarily close to $b$ as input values increase or decrease without bound. The corresponding mathematical notation is $\lim_{x\to\infty} r(x) = b$ or $\lim_{x\to-\infty} r(x) = b$.