Rational Functions Objective: To graph rational functions without a calculator using what we have learned.

Things to look for When graphing rational functions, there are certain things that we need to know. 1)Intercepts 2)Asymptotes 3)Increasing/Decreasing 4)Concave up/Concave down/inflection points 5)Relative extrema

Example 1 Sketch the graph of

Example 1 Sketch the graph of 1) When x = 0, y = ½ 2) This function will equal zero when the numerator is equal to zero. The x-intercepts are (2,0) and (-2,0)

Example 1 Sketch the graph of 3) The vertical asymptotes are the zeros of the denominator which are x = ) The horizontal asymptote is the limit of the function as. Here, since the numerator and denominator are the same degree, the horizontal asymptote is y = 2.

Example 1 Sketch the graph of The graph will never cross a vertical asymptote, but may cross a horizontal asymptote. To see if this happens, we set the equation equal to the asymptote and see if there is an x value that will produce the given y value.

Example 1 Sketch the graph of The graph will never cross a vertical asymptote, but may cross a horizontal asymptote. To see if this happens, we set the equation equal to the asymptote and see if there is an x value that will produce the given y value. This is never true, so it will not cross the asymptote.

Example 1 Sketch the graph of ___+___| ____+____|____-____|___-___ Inc -4 Inc 0 Dec 4 Dec r max sp

Example 1 Sketch the graph of ______+____|______-______|___+____ c up -4 c down 4 c up

Example 1 Here is what the graph looks like. ___+___| ____+____|____-____|___-___ Inc -4 Inc 0 Dec 4 Dec r max ______+____|______-______|___+____ c up -4 c down 4 c up

Example 3 Graph

Example 3 Graph 1)y-intercept (0, -4 2/3 ) 2)X-intercept (4, 0) 3)No horizontal or vertical asymptote.

Example 3 Graph 1)y-intercept (0, -4 2/3 ) 2)X-intercept (4, 0) 3)No horizontal or vertical asymptote. 4) ___-___|___+___ dec 4 inc c.p. (min) 5) ___-___|___-___ c d 4 cd cusp

Example 3 Graph 1)y-intercept (0, -4 2/3 ) 2)X-intercept (4, 0) 3)No horizontal or vertical asymptote. 4) ___-___|___+___ dec 4 inc c.p. (min) 5) ___-___|___-___ c d 4 cd cusp

Example 4 Graph 1)y-intercept (0, 0) 2)x-intercept (-2, 0) and (0, 0) 3)No asymptotes

Example 4 Graph 1)y-intercept (0, 0) 2)x-intercept (-2, 0) and (0, 0) 3)No asymptotes dec inc inc ____-___|___+___|___+___ -1/2 0 min (s.p.) c.p.

Example 4 Graph 1)y-intercept (0, 0) 2)x-intercept (-2, 0) and (0, 0) 3)No asymptotes cu cd cu ____+___|___-___|___+___ 0 1 i.p. i.p.

Example 4 Graph dec inc inc ____-___|___+___|___+___ -1/2 0 min (s.p.) c.p. cu cd cu ____+___|___-___|___+___ 0 1 i.p. i.p.

Homework Section 4.3 Page 264 1, 3, 15 (ignore the instructions), 25, 33, 35