1. 2.7 – Graphs of Rational Functions

2. By then end of today you will learn about……. Rational Functions Transformations of the Reciprocal function Limits and Asymptotes Analyzing Graphs of Rational Functions

3. Rational Functions Rational functions are ratios (or quotients) of polynomial functions Definition: Let f and g be polynomial functions with g(x) ≠ 0. Then the function given by y(x) = f(x) g(x) is a rational function

4. Domain of a Rational Function f(x) = 1 x+ 3 Vertical Asymptote: “As x approaches -3 from the left, the values of f(x) decrease infinitely.” Domain: Use limits to describe its behavior at the vertical asymptotes: “As x approaches -3 from the right, the values of f(x) increase infinitely.”

5. The Basic Reciprocal Function f(x) = 1 x Domain: Range: Continuity: Decreasing on: Symmetry: Bounded? Extrema? Horizontal Asymptote: Vertical Asymptote: End behavior

6. Transforming the Reciprocal Function Use your calculators to graph the following functions. How do they compare to the basic reciprocal function?

7. f(x) = -3 x-1 Transforming the Reciprocal Function Transformations: Domain: Horizontal Asymptote: Vertical Asymptote: Behavior of f(x) at value of x not in domain (or vertical asymptotes):

8. Graphs of Rational Functions End Behavior Asymptote: If numerator degree < denominator degree, then the end behavior asymptote is the horizontal asymptote y=0 If numerator degree = denominator degree, then the end behavior asymptote is the horizontal asymptote: y= leading coefficient of numerator leading coefficient of denominator If numerator degree > denominator degree, the end behavior asymptote is the quotient polynomial function y=q(x), where f(x) = g(x)q(x) + r(x) There is NO horizontal asymptote. Vertical Asymptotes: Occur at the zeros of the denominator X-Intercepts: Occur at the zeros of the numerator Y-intercept: This is the value of f(0), if defined

9. Transforming the Reciprocal Graph f(x) = 2x -1 x + 3 Transformation: Vertical Asymptote: Horizontal Asymptote: Limits to describe f(x) at vertical asymptote:

10. h(x) = x – 1 x 2 – x – 12 X-Intercept: Y-Intercept: Vertical Asymptotes: Horizontal Asymptote: Behavior at Vertical Asymptotes: Domain: Range: Continuity: Increasing Decreasing: Symmetry? Extrema? End behavior:

11. Find the intercepts, asymptotes, use limits to describe the behavior at the vertical asymptotes, and analyze and draw the graph of the rational function X-Intercept: Y-Intercept: Vertical Asymptotes Horizontal Asymptote Behavior at Vertical Asymptotes Domain Range Continuity Decreasing Symmetry? Extrema? End behavior f(x) = x 2 + x -2 x 2 - 9

12. h(x) = x 3 - 2 x + 2 X-Intercept Y-Intercept Vertical Asymptotes End behavior Asymptote Behavior at Vertical Asymptotes Domain Range Continuity Decreasing Increasing: Symmetry? Extrema? End behavior Find the intercepts, analyze, and draw the graph of the rational function

13. Slant Asymptote End Behavior Asymptote: Vertical Asymptote: X-Intercept: Y-Intercept:

14. Don’t forget your homework! pg. 246-247 (4-60, every 4)