Lesson 3 of 4Rational Functions

Unit 2 · Lesson 3 of 4

Sketching Rational Functions

A repeatable seven-step procedure: restrictions, intercepts, asymptotes, sign analysis around every critical value, behaviour near a hole, and then the sketch.

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Key things to remember

1 $x$ -intercepts are the zeros of the simplified numerator; the $y$ -intercept is $f (0)$ when $0$ is in the domain.
2Critical values for the sign chart are the $x$ -intercepts and the vertical asymptotes.
3The sign of $f$ on the interval beside a vertical asymptote tells you whether that branch goes to $+ \infty$ or $- \infty$ .
4Near a hole the curve is smooth and finite; only one point is missing.
5To test whether the curve meets its horizontal asymptote $y = k$ , solve $f (x) = k$ .

Sketching a rational function is not an art project and it is not guesswork. It is a checklist. Work through the same steps in the same order every time and the graph assembles itself: the asymptotes build a frame, the intercepts pin the curve to real points, and a sign analysis tells you which of the regions between the asymptotes each branch lives in. Do the steps out of order and you will end up erasing.

The procedure

Factor the numerator and denominator completely.
State the restrictions (zeros of the original denominator) and simplify if anything cancels.
Find the intercepts. $x$ -intercepts come from zeros of the simplified numerator; the $y$ -intercept is $f (0)$ , if $0$ is in the domain.
Find the asymptotes. Non-cancelling denominator factors give vertical asymptotes; the degree comparison gives the horizontal or oblique asymptote.
Find any holes and their heights from the simplified function.
Sign analysis. Mark every $x$ -intercept and every vertical asymptote on a number line and test the sign of $f$ in each interval.
Sketch, drawing each branch so it approaches its asymptotes and passes through the known points.

VideoA short overview of graphing rational functions by finding the asymptotes first and using them as the frame for the sketch, the same starting point as this checklist.Watch on YouTube

Intercepts

A fraction equals zero only when its numerator equals zero, so the $x$ -intercepts are the zeros of the numerator — but only those that survive as part of the domain. A numerator zero that was also a denominator zero is a hole, not an intercept, so it gets an open circle and no crossing. The $y$ -intercept is simply $f (0)$ , and it exists unless $x = 0$ happens to be a restriction.

Sign analysis: the step people skip

The critical values are the $x$ -intercepts and the vertical asymptotes. Between consecutive critical values the function cannot change sign, because it can only cross the axis at an intercept and can only jump at an asymptote. So pick one convenient test value inside each interval, substitute it into the factored form, and record only the sign of each factor — you never need the actual number. An even number of negative factors gives a positive result.

Fig 1A number line split by the intercepts and vertical asymptotes, with the sign of each factor recorded in every interval.

Interval	x − 1	x + 2	f(x)	Position
$(- \infty, - 2)$	−	−	+	above the x-axis
$(- 2, 1)$	−	+	−	below the x-axis
$(1, \infty)$	+	+	+	above the x-axis

Sign chart for f(x) = (x − 1)/(x + 2), split at the intercept x = 1 and the asymptote x = −2.

The sign chart also tells you the direction the curve takes at each vertical asymptote. Just to the left of $x = - 2$ the function is positive, so that branch climbs to $+ \infty$ ; just to the right it is negative, so that branch plunges to $- \infty$ . You never need to guess which way a branch goes.

Behaviour near a hole

A hole changes almost nothing about the shape. The curve approaches the hole smoothly from both sides at the same height, and you simply leave the single point open. Do not break the curve, do not send it to infinity, and do not draw a vertical line. If you sketch the simplified function and then erase one point, you have drawn the original correctly.

Does the curve cross its horizontal asymptote?

Set the function equal to the asymptote value and solve. If there is a real solution in the domain, the graph really does cross there. If there is none, the curve stays on one side the whole way.

Loading interactive…

Build a numerator and a denominator from factors and watch the asymptotes, holes and branch directions update live.

Three things a rational graph never does

It never touches a vertical asymptote. It never has an output at a restriction. And it never crosses the

x

-axis at a hole — even if the numerator is zero there.

Worked examples

Reveal one step at a time and try to beat it to the next line.

Example 1

Sketch

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

, showing restrictions, intercepts, asymptotes and the sign of each branch.

Example 2

Sketch

f (x) = \frac{x ^{2} - 4}{x ^{2} - x - 6}

, including the hole.

Example 3

Sketch

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

, a case with an oblique asymptote.

Where marks get lost

Letting the curve touch or cross a vertical asymptote. The function has no output there, so the graph must stay strictly on one side of the line.
Testing only one side of a vertical asymptote. The two sides can go the same direction or opposite directions — the sign chart decides, not intuition.
Drawing a hole as a break or a vertical asymptote. A hole is a single missing point on an otherwise smooth curve.
Marking an $x$ -intercept where the numerator is zero but the value is a restriction. That point is a hole, so the graph does not cross there.
Bending the curve away from a horizontal asymptote just because it already crossed it. Crossing in the middle is allowed; the asymptote only governs the far ends.

Flashcards 7

First step in sketching any rational function

How do you find x-intercepts?

How do you find the y-intercept?

Critical values for a sign chart

+ 3 more

Practice

What is the

y

-intercept of

f (x) = \frac{2 x - 6}{x + 3}

The graph of

f (x) = \frac{x ^{2} - 9}{x ^{2} - 2 x - 3}

has a hole. What is its

y

-coordinate?

What are the

x

-intercepts of

f (x) = \frac{x ^{2} - x - 6}{x - 3}

Near the vertical asymptote of

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

, how does the graph behave?

How many vertical asymptotes does

f (x) = \frac{( x - 1 ) ( x + 4 )}{( x - 1 ) ( x + 4 ) ( x - 6 )}

have?

For

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

, is

f

positive or negative on the interval

(- 2, 1)

Order the sketching procedure for a rational function:

1Build a sign chart from the intercepts and vertical asymptotes
2Find the vertical asymptote(s) and the horizontal or oblique asymptote
3Draw the branches through the known points and towards the asymptotes
4Factor the numerator and denominator
5State the restrictions and simplify
6Find the x- and y-intercepts

Match each graph feature to the calculation that finds it.

x-intercept

y-intercept

Vertical asymptote

Height of a hole

Horizontal asymptote

Does the graph of

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

ever cross its horizontal asymptote

y = 0

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