Rational Functions Marvin Marvin Pre-cal Pre-cal

What is Rational Functions? Rational function is a function g that is a quotient of two polynomials. G (x): P (x)/Q (x) Q (x) can ’ t be zero

Vertical Asymptotes The vertical asymptotes of a rational function F(x) = P(x)/Q(x) are found by determining the zeros of Q(x) that are not also zeros of P(x). If P(x) and Q(x) If a is a zero of the denominator but not the numerator, then the line x = a is a vertical asymptote for the graph of the function.

Horizontal Asymptote When the numerator and the denominator of a rational function have the same degree, the line y=a/b is the horizontal asymptote. When the degree of the numerator o fa rational function is less than the degree of the denominator, y=o is the horizontal asymptote

When the degree of the numerator of a rational function is greater tha the degree of the denominator, there is no horizontal asymptote!

Crossing an asymptote! The graph of a rational function never cross a vertical asymptote. The graph of a rational function might cross a horizontal asymptote but does not necessarily do so.

Occurrence of Line As Asymptote of Rational Functions When the degree of the numerator is less than the degree of the denominator---The x-axis is the horizontal asymptote When the numerator and the denominator have the same degree---horizontal asymptote other than the x-axis. When the degree of the numerator is 1 greater than the degree of the denominator--- Oblique asymptote.

How to draw the graph~ 1. Find the real zeros of the denominator. Determine the domain of the function and sketch any vertical asymptotes. 2. Find the horizontal or oblique asymptote, if there is one, and sketch it.

3. Find the zeros of the function. The zeros are found by determining the zeros of the numerator. These are the first coordinates of the x- intercepts of the graph. 4. Find f(0). This gives the y- intercept (0, f(0)), of the function. 5. Find other function values to determine the general shape. Then draw the graph.

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Rational Inequality 1. Find an equivalent inequality with 0 on one side. 2. Change the inequality symbol to an equals sign and solve the related equation, that is, solve f(x) = Find the values of the variable for which the related rational function is not defined. 4. The numbers found in steps (2) and (3) are called critical values. Use the critical values to divide the x-axis into intervals. Then test an x- value from each interval to determine the function ’ s sign in that interval.

5. Select the intervals for which the inequality is satisfied and write interval notation or set-builder notation for the solution set. If the inequality symbol is or , then the solutions to step (2) should be included in the solution set. The x- values found in step (3) are never included in the solution set.