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### Transcript

• 1. Lesson 3.7 Graphs of Rational Functions
• 2. A rational function is a quotient of two polynomial functions. )( )( )( xh xg xf  0)( xh The parent function is , x xf 1 )(  0x
• 3. I III a vertical asymptote x = 0 a horizontal asymptote y = 0
• 4. Example 1. Determine the asymptotes for the graph of f(x) = . 1 34   x x A vertical asymptote Solve x +1 = 0 x = -1 Since f(-1) is undefined, there may be a vertical asymptote at x = -1
• 5. A horizontal asymptote 1 34 )(    x x xf 1 34    x x y Solve for xy(x + 1) = 4x + 3 xy + y = 4x + 3 xy - 4x = 3 - y x(y - 4) = 3 - y 4 3    y y x The rational expression is undefined for y = 4. Thus, the horizontal asymptote is the line y = 4.
• 6. The graph of this function verifies that the lines of y = 4 and x = -1 are asymptotes. f(x) = 1 34   x x
• 7. Here’s the cheat sheet! a) If the degree of the numerator is greater than the degree of the denominator then there is no a horizontal asymptote. 63x 2x y 2   
• 8. b). If the degree of the numerator is less than the degree of the denominator then a horizontal asymptote is y=0 1x 14x y 2   
• 9. c). If the degree of the numerator is equal to the degree of the denominator then a horizontal asymptote is a quotient. 16x 4x y 2 2    4 x 4x y 2 2    a horizontal asymptote
• 10. Example 2 Use the parent graph to graph each function. Describe the transformation(s) that take place. Identify the new location of each asymptote. x 1 f(x)  a. g(x) = To graph g(x) translate the parent graph 2 units to the right. The new vertical asymptote is x = 2. The horizontal asymptote, y = 0, remains unchanged. 2-x 1
• 11. b. h(x) = 3x 1  To graph h(x) , reflect the parent graph over the x-axis, and compress the result horizontally by a factor of 3. This does not affect the vertical asymptote at x = 0. The horizontal asymptote, y = 0, is also unchanged.
• 12. To graph m(x) , stretch the parent function vertically by a factor of 3, and translate the result 1 unit to the right and 2 units down. The new vertical asymptote is x = 1. The horizontal asymptote changes from y = 0 to y = -2. 2 1x 3 m(x)   c.
• 13. A slant asymptote occurs when the degree of the numerator of a rational function is exactly one degree greater then the degree of a denominator.
• 14. Example 3 Determine the slant asymptote for   1x 42x3x xf 2   
• 15. a slant asymptote a vertical asymptote x = 1