Graphing Rational Functions KEY. (x 4) (x + 2) Factor denominator. y = 0 x = 4, x = -2

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1 6 ( 6) Factor numerator 1) f ( ) 8 ( 4) ( + ) Factor denominator n() is of degree: 1 -intercepts: d() is of degree: 6 y 0 4, - Plot the -intercepts. Draw the asymptotes with dotted lines. Then perform a sign analysis to f() VA VA Zero , then ( 6) Shade the ( 4) ( + ) epression for intervals of. + + of f() 01, TESCCC 10/09/1 page 1 of 8

2 3 ( 3) Factor numerator ) f ( ) 9 ( 3)( + 3) Factor denominator n() is of degree: 1 -intercepts: d() is of degree: 1.5 y 0-3, 3 Plot the -intercepts. Draw the asymptotes with dotted lines. Then perform a sign analysis to f() VA Zero VA , then ( 3) Shade the ( 3) ( + 3) epression for intervals of. + + of f() 01, TESCCC 10/09/1 page of 8

3 18 ( 3)( + 3) Factor numerator 3) f ( ) ( + 1)( + 1) Factor denominator n() is of degree: -intercepts: d() is of degree: -3, 3 y -1 Plot the -intercepts. Draw the asymptotes with dotted lines. Then perform a sign analysis to f() Zero VA Zero , then ( 3) Shade the ( + 3) ( + 1 ) ( + 1) Factors that are squared are positive for all of. epression for intervals of. + + of f() 01, TESCCC 10/09/1 page 3 of 8

4 3 10 ( 5)( + ) Factor numerator 4) f ( ) 6 ( 3)( + ) Factor denominator n() is of degree: 1 -intercepts: d() is of degree: 1 5 y Plot the -intercepts. Draw the asymptotes with dotted lines. Then perform a sign analysis to f() RD VA Zero , then ( 5) Shade the ( + ) ( + ) ( 3) The removable factors cancel each other out epression for intervals of of f() 01, TESCCC 10/09/1 page 4 of 8

5 5 ( 5)( + 5) Factor numerator 5) f ( ) 16 ( 4)( + 4) Factor denominator n() is of degree: -intercepts: d() is of degree: -5, 5 y 1 4, -4 Plot the -intercepts. Draw the asymptotes with dotted lines. Then perform a sign analysis to Zero VA VA Zero f() , then ( 5) Shade the ( + 5) ( 4) ( + 4) epression for intervals of of f() 01, TESCCC 10/09/1 page 5 of 8

6 3 3 Factor numerator 6) f ( ) 4 ( )( + ) Factor denominator n() is of degree: 3 -intercepts: d() is of degree: 0, - The line y Plot the -intercepts. Draw the asymptotes with dotted lines. Then perform a sign analysis to f() VA Zero VA , then 3 ( ) Shade the ( + ) epression for intervals of. + + of f() 01, TESCCC 10/09/1 page 6 of 8

7 For each of the rational functions below (#7 through #10), complete the following steps: A) Write down the equations for the vertical asymptotes. B) Using long division or synthetic substitution where appropriate, determine the equations for any slant or end-behavior asymptotes. C) With dotted lines or curves, draw and label the asymptotes on the graphs provided. D) Then, with the aid of a calculator, sketch the graph of each function. 7) 4 f ( ) A) Vertical Asymptote: B) Slant Asymptote: y + 3 8) f ( ) 4 6 A) Vertical Asymptote: 1.5 B) Slant Asymptote: y -(½) + 01, TESCCC 10/09/1 page 7 of 8

8 9) f ( ) + A) Vertical Asymptotes: -, 1 B) Slant Asymptote: y 1 10) f ( ) A) Vertical Asymptote: B) End Behavior Asymptote: y , TESCCC 10/09/1 page 8 of 8

of multiplicity two. The sign of the polynomial is shown in the table below

of multiplicity two. The sign of the polynomial is shown in the table below 161 Precalculus 1 Review 5 Problem 1 Graph the polynomial function P( ) ( ) ( 1). Solution The polynomial is of degree 4 and therefore it is positive to the left of its smallest real root and to the right