M
Math Coach Amy
Amy Ferguson Moncure

IB AA · 3B Sketching Graphs

Intercepts and Asymptotes

Read a rational graph from its equation before you sketch it.

Jump to section:

Key Ideas Easy Medium Hard

VA: denominator = 0

HA: divide the leading coefficients

What to Remember

Vertical asymptote

Set the denominator equal to 0. Cancel a common factor first if there is one. The cancelled factor is a hole, not an asymptote. These items do not cancel.

Horizontal asymptote

When the numerator and the denominator are both degree 1, divide the coefficients of x.

Intercepts

The x-intercept uses numerator = 0. The y-intercept uses x = 0, if that value is allowed.

Sketch

Draw the asymptotes, plot the intercepts, and approach the asymptotes from each side.

Easy

Easy

A simple fraction.

Example
For y = xx − 3, write the equations of the asymptotes and the intercepts.

y = xx − 3

A vertical asymptote is where the denominator is 0 and the numerator is not.

x − 3 = 0

x = 3

For a linear fraction, the horizontal asymptote is the ratio of the x coefficients.

y = 1

x = 0

x = 0

The x-intercept is (0, 0).

At x = 0,

y = 0

The y-intercept is (0, 0).

xy

Answer: VA x = 3, HA y = 1, x-intercept (0, 0), y-intercept (0, 0)

1
For y = x + 1x − 2, write the equations of the asymptotes and the intercepts.
Show Answer

y = x + 1x − 2

A vertical asymptote is where the denominator is 0 and the numerator is not.

x − 2 = 0

x = 2

For a linear fraction, the horizontal asymptote is the ratio of the x coefficients.

y = 1

x + 1 = 0

x = −1

The x-intercept is (−1, 0).

At x = 0,

y = −12

The y-intercept is (0, −12).

xy

Answer: VA x = 2, HA y = 1, x-intercept (−1, 0), y-intercept (0, −12)

2
For y = 4x + 1, write the equations of the asymptotes and the intercepts.
Show Answer

y = 4x + 1

A vertical asymptote is where the denominator is 0 and the numerator is not.

x + 1 = 0

x = −1

For a linear fraction, the horizontal asymptote is the ratio of the x coefficients.

y = 0

The numerator is never 0, so there is no x-intercept.

At x = 0,

y = 4

The y-intercept is (0, 4).

xy

Answer: VA x = −1, HA y = 0, no x-intercept, y-intercept (0, 4)

3
For y = x − 4x + 2, write the equations of the asymptotes and the intercepts.
Show Answer

y = x − 4x + 2

A vertical asymptote is where the denominator is 0 and the numerator is not.

x + 2 = 0

x = −2

For a linear fraction, the horizontal asymptote is the ratio of the x coefficients.

y = 1

x − 4 = 0

x = 4

The x-intercept is (4, 0).

At x = 0,

y = −2

The y-intercept is (0, −2).

xy

Answer: VA x = −2, HA y = 1, x-intercept (4, 0), y-intercept (0, −2)

4
For y = 2x − 5, write the equations of the asymptotes and the intercepts.
Show Answer

y = 2x − 5

A vertical asymptote is where the denominator is 0 and the numerator is not.

x − 5 = 0

x = 5

For a linear fraction, the horizontal asymptote is the ratio of the x coefficients.

y = 0

The numerator is never 0, so there is no x-intercept.

At x = 0,

y = −25

The y-intercept is (0, −25).

xy

Answer: VA x = 5, HA y = 0, no x-intercept, y-intercept (0, −25)

5
For y = 2x − 2x + 3, write the equations of the asymptotes and the intercepts.
Show Answer

y = 2x − 2x + 3

A vertical asymptote is where the denominator is 0 and the numerator is not.

x + 3 = 0

x = −3

For a linear fraction, the horizontal asymptote is the ratio of the x coefficients.

y = 2

2x − 2 = 0

x = 1

The x-intercept is (1, 0).

At x = 0,

y = −23

The y-intercept is (0, −23).

xy

Answer: VA x = −3, HA y = 2, x-intercept (1, 0), y-intercept (0, −23)

Medium

Medium

A coefficient other than 1.

Example
For y = 3x + 6x − 1, write the equations of the asymptotes and the intercepts.

y = 3x + 6x − 1

A vertical asymptote is where the denominator is 0 and the numerator is not.

x − 1 = 0

x = 1

For a linear fraction, the horizontal asymptote is the ratio of the x coefficients.

y = 3

3x + 6 = 0

x = −2

The x-intercept is (−2, 0).

At x = 0,

y = −6

The y-intercept is (0, −6).

xy

Answer: VA x = 1, HA y = 3, x-intercept (−2, 0), y-intercept (0, −6)

1
For y = xx + 4, write the equations of the asymptotes and the intercepts.
Show Answer

y = xx + 4

A vertical asymptote is where the denominator is 0 and the numerator is not.

x + 4 = 0

x = −4

For a linear fraction, the horizontal asymptote is the ratio of the x coefficients.

y = 1

x = 0

x = 0

The x-intercept is (0, 0).

At x = 0,

y = 0

The y-intercept is (0, 0).

xy

Answer: VA x = −4, HA y = 1, x-intercept (0, 0), y-intercept (0, 0)

2
For y = 5x − 5x + 2, write the equations of the asymptotes and the intercepts.
Show Answer

y = 5x − 5x + 2

A vertical asymptote is where the denominator is 0 and the numerator is not.

x + 2 = 0

x = −2

For a linear fraction, the horizontal asymptote is the ratio of the x coefficients.

y = 5

5x − 5 = 0

x = 1

The x-intercept is (1, 0).

At x = 0,

y = −52

The y-intercept is (0, −52).

xy

Answer: VA x = −2, HA y = 5, x-intercept (1, 0), y-intercept (0, −52)

3
For y = 2x + 1x − 3, write the equations of the asymptotes and the intercepts.
Show Answer

y = 2x + 1x − 3

A vertical asymptote is where the denominator is 0 and the numerator is not.

x − 3 = 0

x = 3

For a linear fraction, the horizontal asymptote is the ratio of the x coefficients.

y = 2

2x + 1 = 0

x = −12

The x-intercept is (−12, 0).

At x = 0,

y = −13

The y-intercept is (0, −13).

xy

Answer: VA x = 3, HA y = 2, x-intercept (−12, 0), y-intercept (0, −13)

4
For y = 3x − 12x, write the equations of the asymptotes and the intercepts.
Show Answer

y = 3x − 12x

A vertical asymptote is where the denominator is 0 and the numerator is not.

x = 0

x = 0

For a linear fraction, the horizontal asymptote is the ratio of the x coefficients.

y = 3

3x − 12 = 0

x = 4

The x-intercept is (4, 0).

The denominator is 0 at x = 0, so there is no y-intercept.

xy

Answer: VA x = 0, HA y = 3, x-intercept (4, 0), no y-intercept

5
For y = x − 1x − 4, write the equations of the asymptotes and the intercepts.
Show Answer

y = x − 1x − 4

A vertical asymptote is where the denominator is 0 and the numerator is not.

x − 4 = 0

x = 4

For a linear fraction, the horizontal asymptote is the ratio of the x coefficients.

y = 1

x − 1 = 0

x = 1

The x-intercept is (1, 0).

At x = 0,

y = 14

The y-intercept is (0, 14).

xy

Answer: VA x = 4, HA y = 1, x-intercept (1, 0), y-intercept (0, 14)

Hard

Hard

The intercept may be missing, or the horizontal asymptote is not 1.

Example
For y = 2x + 4x − 2, write the equations of the asymptotes and the intercepts.

y = 2x + 4x − 2

A vertical asymptote is where the denominator is 0 and the numerator is not.

x − 2 = 0

x = 2

For a linear fraction, the horizontal asymptote is the ratio of the x coefficients.

y = 2

2x + 4 = 0

x = −2

The x-intercept is (−2, 0).

At x = 0,

y = −2

The y-intercept is (0, −2).

xy

Answer: VA x = 2, HA y = 2, x-intercept (−2, 0), y-intercept (0, −2)

1
For y = x + 32x − 2, write the equations of the asymptotes and the intercepts.
Show Answer

y = x + 32x − 2

A vertical asymptote is where the denominator is 0 and the numerator is not.

2x − 2 = 0

x = 1

For a linear fraction, the horizontal asymptote is the ratio of the x coefficients.

y = 12

x + 3 = 0

x = −3

The x-intercept is (−3, 0).

At x = 0,

y = −32

The y-intercept is (0, −32).

xy

Answer: VA x = 1, HA y = 12, x-intercept (−3, 0), y-intercept (0, −32)

2
For y = 4x − 8x + 1, write the equations of the asymptotes and the intercepts.
Show Answer

y = 4x − 8x + 1

A vertical asymptote is where the denominator is 0 and the numerator is not.

x + 1 = 0

x = −1

For a linear fraction, the horizontal asymptote is the ratio of the x coefficients.

y = 4

4x − 8 = 0

x = 2

The x-intercept is (2, 0).

At x = 0,

y = −8

The y-intercept is (0, −8).

xy

Answer: VA x = −1, HA y = 4, x-intercept (2, 0), y-intercept (0, −8)

3
For y = x − 6x + 3, write the equations of the asymptotes and the intercepts.
Show Answer

y = x − 6x + 3

A vertical asymptote is where the denominator is 0 and the numerator is not.

x + 3 = 0

x = −3

For a linear fraction, the horizontal asymptote is the ratio of the x coefficients.

y = 1

x − 6 = 0

x = 6

The x-intercept is (6, 0).

At x = 0,

y = −2

The y-intercept is (0, −2).

xy

Answer: VA x = −3, HA y = 1, x-intercept (6, 0), y-intercept (0, −2)

4
For y = 2x − 6x + 5, write the equations of the asymptotes and the intercepts.
Show Answer

y = 2x − 6x + 5

A vertical asymptote is where the denominator is 0 and the numerator is not.

x + 5 = 0

x = −5

For a linear fraction, the horizontal asymptote is the ratio of the x coefficients.

y = 2

2x − 6 = 0

x = 3

The x-intercept is (3, 0).

At x = 0,

y = −65

The y-intercept is (0, −65).

xy

Answer: VA x = −5, HA y = 2, x-intercept (3, 0), y-intercept (0, −65)

5
For y = x + 23x − 6, write the equations of the asymptotes and the intercepts.
Show Answer

y = x + 23x − 6

A vertical asymptote is where the denominator is 0 and the numerator is not.

3x − 6 = 0

x = 2

For a linear fraction, the horizontal asymptote is the ratio of the x coefficients.

y = 13

x + 2 = 0

x = −2

The x-intercept is (−2, 0).

At x = 0,

y = −13

The y-intercept is (0, −13).

xy

Answer: VA x = 2, HA y = 13, x-intercept (−2, 0), y-intercept (0, −13)

Want to go over a graph together?

Book a Session with Amy