KEY FEATURES OF RATIONAL FUNCTIONS WORKSHEET

For each of the following, find the x-intercept, y-intercept, vertical asymptotes and end behaviour.

Problem 1 :

Problem 2 :

Problem 3 :

Problem 4 :

Answers

1. Answer :

y-intercept :

Substitute x = 0 into the given rational function.

y-intercept is (⁵⁄₃, 0).

x-intercept :

Substitute y = 0 into the given rational function.

2(x - 1)(x + 5) = 0

(x - 1)(x + 5) = 0

x - 1 = 0  or  x + 5 = 0

x = 1  or  x = -5

The x-intercepts are (1, 0) and (-5, 0).

Vertical Asymptotes :

Equate the denominator part of the given rational function to zero and solve for x.

(x - 3)(x + 2) = 0

x - 3 = 0  or  x + 2 = 0

x = 3  or  x = -2

The vertical asymptotes are x = 3 and x = -2.

End Behaviour :

Left end behaviour :

When x ---> -∞, y ---> 2.

Right end behaviour :

Whjen x ---> +∞, y ---> 2.

2. Answer :

y-intercept :

Substitute x = 0 into the given rational function.

y-intercept is (6, 0).

x-intercept :

Substitute y = 0 into the given rational function.

(x - 4)(x + 3) = 0

x - 4 = 0  or  x + 3 = 0

x = 4  or  x = -3

The x-intercepts are (4, 0) and (-3, 0).

Vertical Asymptotes :

Equate the denominator part of the given rational function to zero and solve for x.

(x + 1)(x - 2) = 0

x + 1 = 0  or  x - 2 = 0

x = -1  or  x = 2

The vertical asymptotes are x = -1 and x = 2.

End Behaviour :

Left end behaviour :

When x ---> -∞, y ---> 1.

Right end behaviour :

When x ---> +∞, y ---> 1.

3. Answer :

y-intercept :

Substitute x = 0 into the given rational function.

y-intercept is (⁹⁄₅, 0).

x-intercept :

Substitute y = 0 into the given rational function.

3(x + 1)(x + 3)(x - 4) = 0

(x + 1)(x + 3)(x - 4) = 0

x + 1 = 0  or  x + 3 = 0  or  x - 4 = 0

x = -1  or  x = -3  or  x = 4  

The x-intercepts are (-3, 0), (-1, 0) and (4, 0).

Vertical Asymptotes :

Equate the denominator part of the given rational function to zero and solve for x.

(x - 2)(x + 2)(x + 5) = 0

x - 2 = 0  or  x + 2 = 0  or  x + 5 = 0

x = 2  or  x = -2  or  x = -5

The vertical asymptotes are x = -5, x = -2 and x = 2.

End Behaviour :

Left end behaviour :

When x ---> -∞, y ---> 3.

Right end behaviour :

When x ---> +∞, y ---> 3.

4. Answer :

y-intercept :

Substitute x = 0 into the given rational function.

y-intercept is (-4, 0).

x-intercept :

Substitute y = 0 into the given rational function.

(x + 2)(x + 6) = 0

x + 2 = 0  or  x + 6 = 0

x = -2  or  x = -6  

The x-intercepts are (-6, 0) and (-2, 0).

Vertical Asymptotes :

Equate the denominator part of the given rational function to zero and solve for x.

x - 3 = 0

x = 3

The vertical asymptote is x = 3.

End Behaviour :

Left end behaviour :

When x ---> -∞, y ---> -∞.

Right end behaviour :

When x ---> +∞, y ---> +∞.

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