Rational functions: intercepts and holes

Warm-up

Three quick questions from earlier lessons.

Discover

$f(x) = \dfrac{(x - 2)(x + 1)}{x - 2}$ . Simplified, it looks like $x + 1$ . Is $f$ exactly the same function as $x + 1$ ?

Almost: $f$ is not defined at $x = 2$
Yes, exactly the same

Key idea

The numerator and a cancelled factor tell you where the graph meets the axis and where it has a gap.

x-intercepts: zeros of the numerator that are not zeros of the denominator. Hole: a factor that cancels from top and bottom; the hole is at that

x

, and its height is the value of the simplified rule there. y-intercept:

f(0)

.

Worked example

Analyze $f(x) = \dfrac{(x - 3)(x + 4)}{(x - 3)(x - 1)}$ .

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

for

x \ne 3

x − 3 cancels.

Hole at

\left(3, \dfrac{7}{2}\right)

The simplified rule at x = 3: 7/2.

x

-intercept:

x = -4

Numerator zero that remains.

Vertical asymptote:

x = 1

Denominator zero that remains.

Your turn, guided

Your turn: $f(x) = \dfrac{(x + 2)(x - 5)}{x + 2}$ .

Hole at

x = -2

x + 2 cancels: f(x) = x − 5 for x ≠ −2.

The hole is at

(-2, -7)

The simplified rule at x = −2: −2 − 5.

Practice ★

Where does the graph of $f(x) = \dfrac{x - 8}{x + 4}$ cross the $x$ -axis? Give $x$ .
Where does the graph of $f(x) = \dfrac{x - 5}{x + 1}$ cross the $x$ -axis? Give $x$ .
Where does the graph of $f(x) = \dfrac{x - 8}{x + 8}$ cross the $x$ -axis? Give $x$ .
Where does the graph of $f(x) = \dfrac{x - 3}{x + 7}$ cross the $x$ -axis? Give $x$ .

Watch out!

Practice ★★

$f(x) = \dfrac{(x - 5)(x + 6)}{x - 5}$ . At which $x$ does the graph have a hole?
$f(x) = \dfrac{(x - 6)(x + 4)}{x - 6}$ . At which $x$ does the graph have a hole?
$f(x) = \dfrac{(x - 2)(x + 4)}{x - 2}$ . At which $x$ does the graph have a hole?

$f(x) = \dfrac{(x - 1)(x + 3)}{x - 1}$ has a hole at $x = 1$ . What is the $y$ -coordinate of the hole?
$f(x) = \dfrac{(x - 7)(x + 7)}{x - 7}$ has a hole at $x = 7$ . What is the $y$ -coordinate of the hole?

Exit ticket

Three questions to check you have got it.

$f(x) = \dfrac{(x - 1)(x + 3)}{x - 1}$ has a hole at $x = 1$ . What is the $y$ -coordinate of the hole?
$f(x) = \dfrac{(x - 7)(x + 7)}{x - 7}$ has a hole at $x = 7$ . What is the $y$ -coordinate of the hole?
$f(x) = \dfrac{(x - 9)(x + 1)}{x - 9}$ has a hole at $x = 9$ . What is the $y$ -coordinate of the hole?

Summary card

Intercepts and holes

Cancel common factors: each one leaves a hole.
Hole height: the simplified rule at that $x$ .
$x$ -intercepts: remaining zeros of the numerator; $y$ -intercept: $f(0)$ .

Watch out: zeros of the denominator give asymptotes, not intercepts.

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