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Precalculus : Find Intercepts and Asymptotes

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Example Questions

Example Question #11 : Find Intercepts And Asymptotes

Find the slant asymptote for

$y = \frac{2x^2 - 13 x - 7 }{ x - 7 }$ .

Possible Answers:

x = -7

y = 2x - 27

y = 2x + 1

x = 7

This graph does not have a slant asymptote.

Correct answer:

This graph does not have a slant asymptote.

Explanation:

By factoring the numerator, we see that this equation is equivalent to

$y = \frac{(2x+1)(x - 7 )}{x - 7}$ .

That means that we can simplify this equation to y = 2x + 1 .

That means that y = 2x + 1 isn't the slant asymptote, but the equation itself.

x = 7 is definitely an asymptote, but a vertical asymptote, not a slant asymptote.

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Example Question #391 : Pre Calculus

Find the y-intercept of $y=\frac{3-x}{x-3}+3$ , if any.

Possible Answers:

$\infty$

$\textup{The y-intercept does not exist.}$

Correct answer:

Explanation:

Be careful not to confuse this equation with the linear slope-intercept form. The y-intercept of an equation is the y-value when the x-value is zero.

Substitute the value of x=0 into the equation.

$y=\frac{3-0}{0-3}+3$

Simplify the equation.

$y=\frac{3-0}{0-3}+3 = \frac{3}{-3}+3 = -1+3 = 2$

The y-intercept is:

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Example Question #391 : Pre Calculus

Find the horizontal asymptote of the function:

$f(x) = \frac{x^2+6x+9}{x^2+5x+6}$

Possible Answers:

y = 2

y = 0

$y = \frac{1}{2}$

y = 1

y = 3

Correct answer:

y = 1

Explanation:

To find the horizontal asymptote, take the leading term of the numerator and the denominator and divide. In this case:

$f(x) = \frac{x^2}{x^2}$

f(x) = 1

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Example Question #41 : Rational Functions

Find the vertical and horizontal asymptotes of the function

$y=\frac{x^2}{(x+3)(x-3)(x+1)}$

Possible Answers:

Vertical Asymptotes: x=-3,-1,3

Horizontal Asymptote: y=0

Vertical Asymptotes: DNE

Horizontal Asymptote: DNE

Vertical Asymptotes: x=-1,3

Horizontal Asymptote: DNE

Vertical Asymptotes: DNE

Horizontal Asymptote: y=0

Correct answer:

Vertical Asymptotes: x=-3,-1,3

Horizontal Asymptote: y=0

Explanation:

The function

$y=\frac{x^2}{(x+3)(x-3)(x+1)}$

is already in simplified form.

To find the vertical asymptotes, we set the denominator equal to and solve for .

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

yields the vertical asymptotes

x=-3,-1,3

To find the horizontal asymptote, we examine the largest degree of between the numerator and denominator

Note that

(x+3)(x-3)(x+1)=x^3+x^2-9x-9

Because the largest degree of in the numerator is less than the largest degree of in the denominator, or 2<3

we find the horizontal symptote to be

y=0

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Example Question #1 : Asymptotes

Determine the asymptotes, if any: $y=\frac{x^2-2x+1}{x^2-1}$

Possible Answers:

x=-1

$x=-\frac{1}{2}$

$x=\pm1$

x=1

$\textup{There are no asymptotes.}$

Correct answer:

x=-1

Explanation:

Factorize both the numerator and denominator.

$y=\frac{x^2-2x+1}{x^2-1}=\frac{(x-1)(x-1)}{(x+1)(x-1)}$

Notice that one of the binomials will cancel.

The domain of this equation cannot include $x=\pm1$ .

The simplified equation is:

$y=\frac{x-1}{x+1}$

Since the (x-1) term canceled, the (x-1) term will have a hole instead of an asymptote.

Set the denominator equal to zero.

x+1=0

Subtract one from both sides.

x=-1

There will be an asymptote at only: x=-1

The answer is: x=-1

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Example Question #1 : Asymptotes

Where is an asymptote located, if any? $y=\frac{2x-4}{x^2-4}$

Possible Answers:

x=2

$\textup{There is no asymptote.}$

x=-2

y=2

y=-2

Correct answer:

x=-2

Explanation:

Factor the numerator and denominator.

2x-4 = 2(x-2)

x^2-4 = (x+2)(x-2)

Rewrite the equation.

$y=\frac{2x-4}{x^2-4}=\frac{ 2(x-2)}{(x+2)(x-2)}=\frac{2}{x+2}$

Notice that the (x-2) will cancel. This means that the root of (x-2) will be a hole instead of an asymptote.

Set the denominator equal to zero and solve for x.

x+2=0

x=-2

An asymptote is located at: x=-2

The answer is: x=-2

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Precalculus : Find Intercepts and Asymptotes

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #11 : Find Intercepts And Asymptotes

Example Question #391 : Pre Calculus

Example Question #391 : Pre Calculus

Example Question #41 : Rational Functions

Example Question #1 : Asymptotes

Example Question #1 : Asymptotes

All Precalculus Resources

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