SKETCHING THE GRAPH OF RATIONAL FUNCTION WORKSHEET

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i)  Determine the asymptotes.

ii) Find the coordinates of all points where the function crosses its asymptotes.

iii)  Sketch the graph of the function

iv) Discuss the behavior of the function as it approaches its asymptotes. 

Problem 1 :

f(x) = 3x-2

Solution

Problem 2 :

f(x) = 2 - 3x+1

Solution

Problem 3 :

f(x) = (x+3)(x+1)(x-2)

Solution

Problem 4 :

f(x) =x+ 2(x-3)

Solution

Problem 5 :

f(x) =x2-4x2+4

Solution

Answer Key

1)

i) Horizontal asymptote :

 x -axis or y = 0.

ii)  Vertical asymptote :

x = 2

x-intercept :

When y = 0, x + 3 = 0 ==> x = -3

iii)  Sketching the graph :

sketchratfunq1

iv)  End behavior of function :

When x -> 2-, y ->-∞

When x -> 2+, y ->∞

When x -> ∞, y -> 0+

When x -> -∞, y -> 0-

2)  

i) Horizontal asymptote :

x -axis or y = 0.

Vertical asymptote :

At x = -1

ii) y-intercept, at x = 0

(0, -1)

iii)  Sketching the graph :

sketchratfunq2.png

iv) End behavior of the function :

When x -> 2-, y ->-∞

When x -> 2+, y ->∞

When x -> ∞, y -> 0+

When x -> -∞, y -> 0-

3)

i) Horizontal asymptote :

x -axis or y = 0.

ii)  Vertical asymptote :

x + 1 = 0

x = -1

x - 2 = 0

x = 2

ii) y-intercept, at x = 0

(0, -1.5)

iii)  Sketching the graph :

sketchratfunq3.png

iv) End behavior of the function :

When x -> -1-, y ->∞

When x -> -1+, y ->-∞

When x -> 2-, y ->-∞

When x -> 2+, y ->∞

When x -> ∞, y -> 0+

When x -> -∞, y -> 0-

4)

i) Horizontal asymptote :

at y = x.

ii)  Vertical asymptote :

x = 3

When x = 0, y = -0.6

iii)  Sketching the graph :

sketchratfunq4.png

iv) End behavior of the function :

When x -> 3-, y ->-∞

When x -> 3+, y ->∞

When x -> ∞, y ->x+

When x -> -∞, y ->x-

5)

i) Horizontal asymptote :

Equation of horizontal asymptote is y = 1

ii)  Vertical asymptote :

No vertical asymptote. 

iii)  Sketching the graph :

sketchratfunq5.png

iv) End behavior :

When x -> ∞, y ->1-

When x -> -∞, y ->1-

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