C2.3 Sketch rational functions from key features

Ontario Grade 12 Mathematics

Polynomial and Rational Functions

Use intercepts, asymptotes, holes, and a few points to build a reliable sketch

A rational function is a quotient of polynomial expressions. Its graph can have gaps or branches that approach lines without touching them. To sketch one, find its key features first. Then use those features to place branches and selected points. This lesson focuses on reading and using those features, not on drawing every possible point.

What you will learn

Identify the domain restrictions of a rational function.
Find intercepts, vertical asymptotes, horizontal asymptotes, and holes.
Use key features and selected points to sketch a rational function.
Check that a sketch matches the function's important features.

1. Prerequisite bridge: factors, intercepts, and restrictions

A factor is an expression multiplied by another expression. For example,

x-3

is a factor of

(x-3)(x+2)

. Factoring helps reveal where a rational function is zero and where its denominator is zero.

The denominator cannot equal zero. Any value that makes it zero is excluded from the domain, which is the set of allowed input values. Keep this restriction even if a common factor later cancels.

An

x

-intercept is where the graph crosses or touches the

x

-axis. Its

y

-value is zero. A

y

-intercept is where the graph meets the

y

-axis, found by setting

x=0

, if the function is defined there.

Factor the numerator and denominator when possible.
Record denominator restrictions before simplifying.
A zero of the numerator can give an $x$ -intercept, unless that input is excluded.

2. What the key features show

A vertical asymptote is a vertical line that a graph approaches as its input nears a particular value. A denominator zero that remains after common factors are cancelled gives a vertical asymptote.

A hole is a missing point in the graph. It occurs when the numerator and denominator share a factor that cancels. The cancelled input is still excluded, and the reduced expression gives the missing point's

y

-coordinate.

A horizontal asymptote describes the height the graph approaches far to the left or right. For a rational function with the same degree in its numerator and denominator, divide the leading coefficients to find this asymptote. The degree is the greatest exponent of the variable in a polynomial.

These features do not always tell the entire shape by themselves. A few carefully chosen points on each side of a vertical asymptote help show where the branches lie. Values from the function are called outputs.

y=\frac{a}{b}

A remaining denominator zero gives a vertical asymptote.
A cancelled factor gives a hole, not a vertical asymptote.
For equal numerator and denominator degrees, the horizontal asymptote is the ratio of leading coefficients.

3. A repeatable sketching plan

Start by factoring and listing every value excluded by the original denominator. Cancel common factors to make the remaining structure easier to read, but do not erase the restrictions.

Find the

x

-intercepts from the zeros of the reduced numerator that are still in the domain. Find the

y

-intercept by evaluating the function at zero when zero is allowed. Identify vertical asymptotes from the remaining denominator factors and holes from cancelled factors.

Find the horizontal asymptote when the degrees are equal. Plot the intercepts, asymptotes, and holes. A hole is shown as an open circle. Choose test inputs in the intervals separated by vertical asymptotes. Their outputs indicate whether each branch is above or below the horizontal axis and help guide the sketch.

Draw smooth branches that pass through the plotted points and approach the asymptotes in the appropriate regions. An asymptote is a guide for the graph's behavior, not an intercept that the function must cross.

Keep original domain restrictions throughout the work.
Use a table or selected points to locate branches between asymptotes.
Mark a hole with an open circle at its calculated coordinates.

4. Guided example

Consider the function below. It has a shared factor, so its graph has a hole as well as a vertical asymptote. We will identify the features before describing the sketch.

The original denominator is zero at

x=1

and

x=3

, so both inputs are excluded. Cancelling the shared factor makes it easier to find the remaining graph shape, but both exclusions still matter.

f(x)=\frac{(x+2)(x-1)}{(x-1)(x-3)}

The cancelled input identifies the hole.
The denominator factor left after cancellation identifies the vertical asymptote.

5. Applying the features and checking the sketch

The reduced expression is

(x+2)/(x-3)

, with

x=1

and

x=3

excluded. The value at the cancelled input is found from the reduced expression: at

x=1

, the output is

-3/2

. Place an open circle at

(1,-3/2)

.

The reduced numerator is zero at

x=-2

, giving the

x

-intercept

(-2,0)

. At

x=0

, the output is

-2/3

, giving the

y

-intercept

(0,-2/3)

. The remaining denominator is zero at

x=3

, so draw a vertical asymptote there.

The numerator and denominator of the reduced expression have the same degree and leading coefficient, so the horizontal asymptote is

y=1

. To see how the branches sit, evaluate one input in each region. At

x=2

, the output is

-4

, below the horizontal axis. At

x=4

, it is

6

, above the horizontal asymptote. These values guide the branches on either side of

x=3

.

The sketch should show a left-side branch passing through the two intercepts and approaching the vertical asymptote as

x

nears

3

from below. It also has a gap at

(1,-3/2)

. The right-side branch is above the horizontal asymptote at the test input

x=4

. The table summarizes the features; selected points guide the shape but do not replace the asymptotes.

A feature table helps prevent missing a restriction or mislabelling a hole.
Check each branch against both the asymptotes and the selected outputs.

Feature summary for the example

Feature	Result	Meaning for the sketch
Domain restrictions	$x\ne 1,\ x\ne 3$	No graph point at either input
Hole	$(1,-3/2)$	Show an open circle
$x$ -intercept	$(-2,0)$	Graph meets the horizontal axis
$y$ -intercept	$(0,-2/3)$	Graph meets the vertical axis
Vertical asymptote	$x=3$	Branches approach this vertical line
Horizontal asymptote	$y=1$	Branches approach this horizontal line at the ends

Worked example

Find the features and describe a sketch

Sketch

f(x)=\frac{(x+2)(x-1)}{(x-1)(x-3)}

using its key features.

Record restrictions
The original denominator is zero at $x=1$ and $x=3$ . The function is undefined at both inputs, so neither value is in the domain.
$x\ne 1,\quad x\ne 3$
Simplify and locate the hole
Cancel the shared factor to find the simpler expression for the graph's values where it is defined. The cancelled input remains excluded. Substituting $x=1$ into the reduced expression gives the hole's height.
$\frac{(x+2)(x-1)}{(x-1)(x-3)}=\frac{x+2}{x-3},\quad f(1)=-\frac{3}{2}$
Find the intercepts
Set the reduced numerator to zero for the $x$ -intercept. Then substitute zero for $x$ to find the $y$ -intercept. Both inputs are allowed.
$(-2,0),\quad \left(0,-\frac{2}{3}\right)$
Find the asymptotes
The remaining denominator is zero at $x=3$ , giving the vertical asymptote. The reduced numerator and denominator have equal degree and leading coefficients of $1$ , giving the horizontal asymptote.
$x=3,\quad y=1$
Use test points
Evaluate one input in each interval split by the vertical asymptote. These values show the branch positions and support a sketch through the known features.
$f(0)=-\frac{2}{3},\quad f(2)=-4,\quad f(4)=6$

Answer: Draw a vertical asymptote at

x=3

and a horizontal asymptote at

y=1

. Plot the intercepts

(-2,0)

and

(0,-2/3)

. Mark an open circle at

(1,-3/2)

. Sketch branches that fit these points and approach the asymptotes.

Check: The input

x=1

is still excluded after cancellation, so the graph has a hole there. The input

x=3

remains in the denominator, so it gives a vertical asymptote.

Common mistakes and how to avoid them

Removing a cancelled input from the domain restriction.

Correction: Restrictions come from the original denominator. A cancelled input still creates a hole.

Calling every original denominator zero a vertical asymptote.

Correction: Cancel common factors first. A cancelled factor creates a hole; a remaining denominator zero gives a vertical asymptote.

Drawing an asymptote as a line the graph must cross or touch.

Correction: An asymptote describes a line the graph approaches. Do not assume the graph intersects it.

Drawing branches from asymptotes alone.

Correction: Add intercepts and selected points so the position of each branch is clear.

Lesson summary

Factor the numerator and denominator, and keep all restrictions from the original denominator.
Use the reduced expression to find intercepts, holes, and vertical asymptotes.
For equal degrees, use the ratio of leading coefficients for the horizontal asymptote.
Plot key features and selected points, then draw branches that fit them.

Check your understanding

Question 1

For

g(x)=\frac{(x-4)(x+1)}{(x-4)(x+2)}

, what feature occurs at

x=4

?

A vertical asymptote
A hole
An $x$ -intercept
A horizontal asymptote

Show answer and explanation

A hole

The factor

x-4

cancels, but

x=4

remains excluded. The reduced expression gives the missing point's height, so the graph has a hole.

Question 2

For

h(x)=\frac{3x+1}{x-2}

, what is the horizontal asymptote?

$y=3$
$y=1/3$
$x=2$
$y=0$

Show answer and explanation

y=3

The numerator and denominator have equal degree. Their leading coefficients are

3

and

1

, so the horizontal asymptote is

y=3

.

Question 3

For

p(x)=\frac{x+5}{x-1}

, which input gives the vertical asymptote?

$x=-5$
$x=0$
$x=1$
$y=1$

Show answer and explanation

x=1

The denominator is zero at

x=1

, and its factor does not cancel. Therefore,

x=1

is the vertical asymptote.

Key terms

Rational function: A function written as one polynomial divided by another polynomial, where the denominator is not zero.
Domain: The set of input values for which a function is defined.
Hole: A missing point caused by a common factor that cancels, while its input remains excluded.
Vertical asymptote: A vertical line that a graph approaches near a particular input.
Horizontal asymptote: A horizontal line that a graph approaches far to the left or right.
Intercept: A point where a graph meets one of the coordinate axes.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation C2.3. It is a study resource, not an official curriculum publication.

Before publication, the draft is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability, and then requires administrator approval. Errors can still occur, so corrections are welcomed.

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C2.3 · Sketch rational functions from key features

Ontario Grade 12 Mathematics

Use intercepts, asymptotes, holes, and a few points to build a reliable sketch

What you will learn

1. Prerequisite bridge: factors, intercepts, and restrictions

2. What the key features show

3. A repeatable sketching plan

4. Guided example

5. Applying the features and checking the sketch

Feature summary for the example

Find the features and describe a sketch

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MHF4U

About this lesson and its review