GRAPHING RATIONAL FUNCTIONS WORKSHEET

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Identify the points of discontinuity, holes, vertical asymptotes, x – intercepts, horizontal asymptote, and domain of each. Then sketch the graph.

Problem 1 :

f(x) = (x + 1)/(x – 2)

Solution

Problem 2 :

f(x) = (x2 + 2x – 8)/(-x2 + 6x – 8)

Solution

Problem 3 :

For each function

a) f(x) = 2/(x - 3)

b) f(x) = (x - 2) / (3x + 4)

c) (x) = (x - 3)/(2x - 6)

i) determine the domain, intercepts, asymptotes, and positive/negative intervals

ii) use these characteristics to sketch the graph of the function

iii) describe where the function is increasing or decreasing

Solution

Answer Key

1)  discontinuous at x = 2.

No hole.

Vertical asymptote at x = 2

x – intercepts is -1.

Horizontal asymptote is y = 1

All real values except  2

2)  Discontinuous at x = 2 and x = 4.

Hole at x = 2.

Vertical asymptote at x = 4.

x – intercepts is -4.

Horizontal asymptote is y = -1.

All real values except 2 and 4.

3) Domain is all real values except 3.

i) no x-intercept, y intercept is y = -2/3

  • Vertical asymptote is at x = 3
  • x-axis is the horizontal asymptote.
  • When x ∈ (-∞, 3), y is negative
  • When x ∈ (3, ∞), y is positive.

ii) Characteristics of graph :

When x --> -∞ then y-->-∞

When x --> ∞ then y--> ∞

iii) describe where the function is increasing or decreasing :

The function is decreasing in the entire domain, when x ∈ (-∞, 3) and x ∈ (3, ∞).

graphing-rational-function-increase-decrease-q1

Identify the points of discontinuity, holes, vertical asymptotes, x – intercepts, horizontal asymptote, and domain of each. Then sketch the graph.

Problem 1 :

f(x) = (x2 - 7x + 12)/(-2x2 - 2x + 24)

Solution

Problem 2 :

 f(x) = (3x2 - 3x - 18)/(x2 - 4)

Solution

Problem 3 :

f(x) = (x2 + 6x + 8)/(x2 + 3x - 4)

Solution

Problem 4 :

f(x) = (2x + 6)/(x2 + 5x + 6)

Solution

Problem 5 :

f(x) = (-2x - 6)/x

Solution

Problem 6 :

f(x) = (-3x + 12)/(x – 2)

Solution

1)  Discontinuous at x = 3 and x = -4.

Hole at x = 3.

Vertical asymptote at x = -4.

x – intercepts is 4.

Horizontal asymptote is y = -1/2.

All real values except 3 and -4.

2)  discontinuous at  x = 2 and x = -2.

Hole at x = -2.

Vertical asymptotes at x = 2.

x – intercepts is 3.

Horizontal asymptote is y = 3.

All real values except -2 and 2.

3)  Discontinuous at x = -4 and x = 1.

Hole at x = -4.

Vertical asymptotes at x = 1.

x – intercepts is -2.

Horizontal asymptote is y = 1.

All real values except -4 and 1

4)  Discontinuous at x = -2 and x = -3.

Hole at x = -3.

Vertical asymptotes at x = -2.

x – intercepts is none.

Horizontal asymptote is y = 0

All real values except -3 and -2.

5)  Discontinuous at x = 0.

No hole

Vertical asymptotes at x = 0.

x – intercepts is -3.

Horizontal asymptote is y = -2.

All real values except 0

6)  discontinuous at x = 2.

No hole

vertical asymptote at x = 2.

x – intercepts is 4.

horizontal asymptote is y = -3.

all real values except 2

Problem 1 :

 f(x) = -3/x

Solution

Problem 2 :

f(x) = (x3 + x2 – 12x)/(x3 – x2 – 6x)

Solution

Problem 3 :

The graph of the rational function f(x) = x2/(x2 - x + 2) intersects its asymptote y = 1. Find the point of intersection.

Solution

Problem 4 :

Match the equation of each rational function with its corresponding graph. Use key features such as intercepts and asymptotes to help you.

a) f(x) = (x + 4) / (x + 8)

b) f(x) = (x2 + 12x + 32) / (x + 8)

c) f(x) = (x2 + 12x + 32) / (x2 + 10x + 16)

d) f(x) = (x2 + 5x + 4) / (x2 + 10x + 16)

rational-function-and-graph-q1

Solution

Answer Key

1)  

Discontinuous at x = 0.

No hole

vertical asymptotes at x = 0.

No x – intercept.

Horizontal asymptote is y = 0

All real values except 0.

2) 

Discontinuous at x = 0, x = -2 and x = 3.

Holes are at x = 0, x = 3.

Vertical asymptote at x = -2.

x – intercepts is -4.

Horizontal asymptote is y = 1.

All real values except 0, -2 and 3.

3)  the point of intersection is at (2, 1)

4)

  • a) f(x) = (x + 4) / (x + 8) ==> graph C
  • b) f(x) = (x2 + 12x + 32) / (x + 8) ==> graph B
  • c) f(x) = (x2 + 12x + 32) / (x2 + 10x + 16) ==> graph D
  • d) f(x) = (x2 + 5x + 4) / (x2 + 10x + 16) ==> graph A

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