AP Precalculus

~ 1.8-1.10: More on Rational Functions ~

1.8 Course Objectives

The real zeros of a rational function correspond to the real zeros of the numerator for such values in its domain.
The real zeros of both polynomial functions of a rational function r are endpoints or asymptotes for intervals satisfying the rational function inequalities r(x) ≥ 0 or r(x) ≤ 0.

If the value a is a real zero of the polynomial function in the denominator of a rational function and is not also a real zero of the polynomial function in the numerator, then the graph of the rational function has a vertical asymptote at x = a.
A vertical asymptote also occurs at x = a if the multiplicity of a as a real zero in the denominator is greater than its multiplicity as a real zero in the numerator.
Near a vertical asymptote, x = a, of a rational function, the values of the polynomial function in the denominator are arbitrarily close to zero, so the values of the rational function r increase or decrease without bound. Know the corresponding mathematical notation.

If the multiplicity of a real zero in the numerator is greater than or equal to its multiplicity in the denominator, then the graph of the rational function has a hole at the corresponding input value.
If the graph of a rational function r has a hole at x = c, then the location of the hole can be determined by examining the output values corresponding to input values sufficiently close to c.
If input values sufficiently close to c correspond to output values arbitrarily close to L, then the hole is located at the point with coordinates (c, L). Know the corresponding mathematical notation.

The video for AP Precalculus, 1.6-1.7: End Behavior of Polynomials and Rational Functions will come out on Thursday, September 21, 2023.