1.  Find the horizontal and vertical asymptotes of the following rational functions 1. (2x) / (3x 2 +1) 2. (2x 2 ) / (x 2 – 1) Note: Vertical asymptotes- set denominator equal to zero and solve. Look at your chart for horizontal asymptotes.

7. Essential Question: How do you sketch graphs of rational functions? Standard: MM4A4. Students will investigate functions. a. Compare and contrast properties of functions within and across the following types: linear, quadratic, polynomial, power, rational, exponential, logarithmic, trigonometric, and piecewise.

8. Vertical Asymptotes of Rational Functions Review If the graph of a rational function has vertical asymptotes, they can be located by using the following theorem. Locating Vertical Asymptotes If is a rational function in which p(x) and q(x) have no common factors and a is a zero of q(x), the denominator, then x  a is a vertical asymptote of the graph of f. Locating Vertical Asymptotes If is a rational function in which p(x) and q(x) have no common factors and a is a zero of q(x), the denominator, then x  a is a vertical asymptote of the graph of f.

9. Horizontal Asymptotes of Rational Functions If the graph of a rational function has a horizontal asymptote, it can be located by using the following theorem. Locating Horizontal Asymptotes Let f be the rational function given by The degree of the numerator is n. The degree of the denominator is m. 1.If n  m, the x-axis is the horizontal asymptote of the graph of f. 2.If n  m, the line y  is the horizontal asymptote of the graph of f. 3.If n  m, the graph of f has no horizontal asymptote. Locating Horizontal Asymptotes Let f be the rational function given by The degree of the numerator is n. The degree of the denominator is m. 1.If n  m, the x-axis is the horizontal asymptote of the graph of f. 2.If n  m, the line y  is the horizontal asymptote of the graph of f. 3.If n  m, the graph of f has no horizontal asymptote. Graphs of Rational functions

10. 3.6: Rational Functions and Their Graphs Strategy for Graphing a Rational Function Suppose that where p(x) and q(x) are polynomial functions with no common factors. 1. Determine whether the graph of f has symmetry. f (  x)  f (x): y-axis symmetry f (  x)  f (x): origin symmetry 2. Find the y-intercept (if there is one) by evaluating f (0). 3. Find the x-intercepts (if there are any) by solving the equation p(x)  0. 4. Find any vertical asymptote(s) by solving the equation q (x)  0. 5. Find the horizontal asymptote (if there is one) using the rule for determining the horizontal asymptote of a rational function. 6. Plot at least one point between and beyond each x-intercept and vertical asymptote. 7. Use the information obtained previously to graph the function between and beyond the vertical asymptotes. Strategy for Graphing a Rational Function Suppose that where p(x) and q(x) are polynomial functions with no common factors. 1. Determine whether the graph of f has symmetry. f (  x)  f (x): y-axis symmetry f (  x)  f (x): origin symmetry 2. Find the y-intercept (if there is one) by evaluating f (0). 3. Find the x-intercepts (if there are any) by solving the equation p(x)  0. 4. Find any vertical asymptote(s) by solving the equation q (x)  0. 5. Find the horizontal asymptote (if there is one) using the rule for determining the horizontal asymptote of a rational function. 6. Plot at least one point between and beyond each x-intercept and vertical asymptote. 7. Use the information obtained previously to graph the function between and beyond the vertical asymptotes.

11. EXAMPLE:Graphing a Rational Function Step 4Find the vertical asymptotes: Set q(x)  0. x 2  4  0 Set the denominator equal to zero. x 2  4 x   2 Vertical asymptotes: x  2 and x  2. Solution mor e Step 3Find the x-intercept: 3x 2  0, so x  0: x-intercept is 0. Step 1Determine symmetry: f (  x)    f (x): Symmetric with respect to the y-axis. Step 2Find the y-intercept: f (0)  0: y-intercept is 0.

12. EXAMPLE:Graphing a Rational Function Solution The figure shows these points, the y-intercept, the x-intercept, and the asymptotes. x 33 11 134 f(x) 11 11 4 mor e Step 6Plot points between and beyond the x-intercept and the vertical asymptotes. With an x-intercept at 0 and vertical asymptotes at x  2 and x  2, we evaluate the function at  3,  1, 1, 3, and 4. Step 5Find the horizontal asymptote: y  3 / 1  3. -5-4-3-212345 7 6 5 4 3 1 2 -2 -3 Vertical asymptote: x = 2 Vertical asymptote: x = -2 Horizontal asymptote: y = 3 x-intercept and y-intercept 3.6: Rational Functions and Their Graphs

13. EXAMPLE:Graphing a Rational Function Solution Step 7Graph the function. The graph of f (x)  2x is shown in the figure. The y-axis symmetry is now obvious. -5-4-3-212345 7 6 5 4 3 1 2 -2 -3 x = -2 y = 3 x = 2 -5-4-3-212345 7 6 5 4 3 1 2 -2 -3 -5-4-3-212345 7 6 5 4 3 1 2 -2 -3 Vertical asymptote: x = 2 Vertical asymptote: x = -2 Horizontal asymptote: y = 3 x-intercept and y-intercept 3.6: Rational Functions and Their Graphs

14. EXAMPLE:Finding the Slant Asymptote of a Rational Function Find the slant asymptotes of f (x)  Solution Because the degree of the numerator, 2, is exactly one more than the degree of the denominator, 1, the graph of f has a slant asymptote. To find the equation of the slant asymptote, divide x  3 into x 2  4x  5: 2 1  4  5 1 3  3 1  1  8 3 Remainder 3.6: Rational Functions and Their Graphs mor e

15. EXAMPLE:Finding the Slant Asymptote of a Rational Function Find the slant asymptotes of f (x)  Solution The equation of the slant asymptote is y  x  1. Using our strategy for graphing rational functions, the graph of f (x)  is shown. -245678321 7 6 5 4 3 1 2 -3 -2 Vertical asymptote: x = 3 Vertical asymptote: x = 3 Slant asymptote: y = x - 1 Slant asymptote: y = x - 1 3.6: Rational Functions and Their Graphs

16. Home work: P153 # 13-16, 23, 31 (no graphing) P 161 # 9, 10