C2.1 Graph reciprocals of linear and quadratic functions

Ontario Grade 12 Mathematics

Polynomial and Rational Functions

Use zeros, signs, and selected values to sketch reciprocal graphs

A reciprocal function is formed by taking one divided by the output of another function. Its graph is closely linked to the original graph, but it is undefined wherever the original output is zero. In this lesson, you will use zeros, signs, asymptotes, and selected values to sketch reciprocals of linear and quadratic functions.

What you will learn

Explain how the graph of a reciprocal function relates to its original linear or quadratic function.
Identify excluded inputs and asymptotes when graphing reciprocals.
Use signs and selected values to sketch reciprocals of linear and quadratic functions.

1. Prerequisite bridge: reciprocals, zeros, and signs

The reciprocal of a nonzero number is one divided by that number. For example, the reciprocal of

4

is

1/4

, and the reciprocal of

-2

is

-1/2

. Zero has no reciprocal because division by zero is undefined.

A function assigns an output to each allowed input. A zero of a function is an input that makes its output equal to zero. For example, if

f(3)=0

, then

3

is a zero of

f

. Zeros matter because the reciprocal function divides one by the original output.

For a function

f

, its reciprocal function is

g(x)=1/f(x)

. It is defined only where

f(x)

is not zero. At each allowed input, the input stays the same and the output becomes its reciprocal.

g(x)=\frac{1}{f(x)}

Find the zeros of the original function before sketching its reciprocal.
At each zero of the original function, the reciprocal is undefined.
Positive outputs stay positive and negative outputs stay negative when reciprocated.

2. Reciprocal of a linear function

A linear function has a straight-line graph. A non-horizontal linear function can be written as

f(x)=mx+b

, where

m

is its slope and

b

is its vertical intercept. Its reciprocal is

g(x)=1/(mx+b)

.

First find the zero of the line. At that input, the reciprocal has a vertical asymptote: a vertical line that the graph approaches near an input where it is undefined. On either side of the zero, use the sign of the line to decide whether the reciprocal branch is above or below the

x

-axis.

For large positive or negative inputs, the magnitude of a non-horizontal line's output grows. Its reciprocal gets closer to zero. So

y=0

is a horizontal asymptote, a horizontal line the graph approaches as inputs become very large or very negative.

For example,

f(x)=x-2

is negative when

x<2

and positive when

x>2

. Its reciprocal is below the

x

-axis to the left of

x=2

, above the axis to the right, and undefined at

x=2

. A few points can be found by taking the reciprocal of each nonzero line output.

g(x)=\frac{1}{mx+b}

The zero of a non-horizontal linear function gives the reciprocal's vertical asymptote.
The reciprocal of a non-horizontal linear function approaches zero for large positive and negative inputs.
The reciprocal graph does not cross $y=0$ , because its numerator is one.

3. Reciprocal of a quadratic function

A quadratic function has a parabolic graph. It can be written in factored form as

f(x)=a(x-r_1)(x-r_2)

when it has two real zeros, or in vertex form as

f(x)=a(x-h)^2+k

. The values

r_1

and

r_2

name the zeros;

(h,k)

is the vertex, or turning point, of the parabola.

The reciprocal is

g(x)=1/f(x)

. Each real zero of the quadratic is excluded from the reciprocal's domain. The domain is the set of inputs for which a function is defined. A reciprocal graph has a vertical asymptote at each real zero. If the quadratic has no real zeros, it has no vertical asymptotes caused by zeros of its denominator.

Use the parabola's sign to place the reciprocal branches. Where the quadratic is above the

x

-axis, its reciprocal is positive. Where it is below the axis, its reciprocal is negative. For a quadratic with two distinct zeros, check the intervals between and outside the zeros to determine the sign on each part.

For large positive or negative inputs, the magnitude of a quadratic output grows, so its reciprocal approaches zero. Thus

y=0

is a horizontal asymptote. Selected values show where branches lie and how they approach the asymptotes.

g(x)=\frac{1}{a(x-r_1)(x-r_2)}

Real zeros of a quadratic become vertical asymptotes of its reciprocal.
The reciprocal has the same sign as the quadratic wherever it is defined.
The reciprocal of a quadratic approaches $y=0$ at both far ends of the graph.

4. A reliable sketching process

Start with the original linear or quadratic function. Identify its zeros, determine where it is positive or negative, and choose a few inputs that are not zeros. Calculate the original outputs, then take their reciprocals.

Plot the resulting points and draw vertical asymptotes at excluded inputs. Include the horizontal asymptote when the reciprocal approaches

y=0

. Draw branches that match the signs and selected points. The reciprocal is undefined at a vertical asymptote, so the graph cannot include a point on that line.

A reciprocal point is not found by reflecting the original point across an axis. If an original point is

(x,y)

with

y\ne0

, its corresponding reciprocal point is

(x,1/y)

. The input stays the same; only the output changes.

(x,y)\mapsto\left(x,\frac{1}{y}\right)

Keep each input fixed and take the reciprocal of its nonzero output.
Use asymptotes as guides, not as parts of the graph.
Check that every branch agrees with the sign of the original function.

Selected points for the worked quadratic

Input $x$	Original output $f(x)=x^2-4$	Reciprocal output $g(x)=1/f(x)$
$-3$	$5$	$1/5$
$-1$	$-3$	$-1/3$
$0$	$-4$	$-1/4$
$1$	$-3$	$-1/3$
$3$	$5$	$1/5$

Worked example

Sketch a reciprocal quadratic

Sketch the reciprocal of

f(x)=x^2-4

. Identify its domain restrictions, asymptotes, signs, and several points.

Find the zeros
Factor the quadratic to find where its output is zero. Those inputs cannot be used in the reciprocal, so they also locate its vertical asymptotes.
$x^2-4=(x-2)(x+2)$
State restrictions and asymptotes
The quadratic is zero at $x=-2$ and $x=2$ . The reciprocal is undefined at both inputs. Since the quadratic's outputs grow in magnitude at both ends, the reciprocal approaches zero there.
$x\ne -2,\quad x\ne 2$
Determine signs
The parabola opens upward. It is above the $x$ -axis outside its zeros and below the axis between them. The reciprocal keeps those signs, so its outer branches are positive and its middle branch is negative. f(x)>0 for x<-2 or x>2, f(x)<0 for -2<x<2
Calculate useful points
Choose inputs away from the zeros. Evaluate the quadratic, then replace each nonzero output with its reciprocal. These points anchor the sketch on different intervals.
$f(0)=-4,\quad g(0)=-\frac14;\qquad f(3)=5,\quad g(3)=\frac15$
Draw the branches
Draw vertical guides at $x=-2$ and $x=2$ , and a horizontal guide at $y=0$ . Plot the calculated points and sketch three branches: positive outside the vertical guides and negative between them. Each branch approaches the nearby guides without touching them.

Answer: The reciprocal is

g(x)=1/(x^2-4)

, with domain

x\ne -2,2

. It has vertical asymptotes

x=-2

and

x=2

, and horizontal asymptote

y=0

. Its outer branches are above the

x

-axis, and its middle branch is below.

Check: At

x=0

, the original output is

-4

, so the reciprocal point is

(0,-1/4)

. This confirms that the middle branch is negative.

Common mistakes and how to avoid them

Including a zero of the original function in the reciprocal's domain.

Correction: At a zero, the reciprocal would require division by zero. Exclude that input and show a vertical asymptote there.

Taking the reciprocal of the input instead of the output.

Correction: Keep the input unchanged. If the original point is

(x,y)

, the reciprocal point is

(x,1/y)

, provided

y\ne0

.

Drawing every reciprocal branch above the

x

-axis.

Correction: A reciprocal keeps the sign of the original output. Check where the line or parabola is positive and negative.

Drawing an asymptote as part of the graph.

Correction: An asymptote is a guide that the graph approaches. A vertical asymptote at a zero is not in the reciprocal's domain.

Lesson summary

The reciprocal of a function is found by taking one divided by each nonzero output.
Zeros of the original function are excluded inputs and give vertical asymptotes.
The reciprocal has the same sign as the original function wherever it is defined.
For non-horizontal linear and quadratic functions, the reciprocal approaches $y=0$ at large positive and negative inputs.
Use asymptotes, signs, and calculated points together to make a reliable sketch.

Check your understanding

Question 1

The linear function

f(x)=3x+6

has a zero at

x=-2

. What is true about its reciprocal there?

It is defined and equals $-2$ .
It is undefined and has a vertical asymptote at $x=-2$ .
It has a horizontal asymptote at $x=-2$ .
It crosses the $x$ -axis at $x=-2$ .

Show answer and explanation

It is undefined and has a vertical asymptote at

x=-2

.

At

x=-2

, the linear function equals zero, so taking its reciprocal would require division by zero. The reciprocal is undefined there, and the zero gives a vertical asymptote.

Question 2

If a quadratic has a negative output at an allowed input, what is the sign of its reciprocal output?

Positive
Negative
Zero
Undefined

Show answer and explanation

Negative

The reciprocal of a negative nonzero number is negative. The reciprocal keeps the original function's sign.

Question 3

The graph of a reciprocal quadratic has a horizontal asymptote

y=0

. What does this tell you about its outputs for inputs with very large magnitude?

They get closer to zero.
They become exactly zero at a finite input.
They approach the quadratic's vertex.
They grow without bound.

Show answer and explanation

They get closer to zero.

A horizontal asymptote at

y=0

means the reciprocal outputs get closer to zero as inputs become very large in magnitude. The reciprocal is not zero at any input where it is defined.

Key terms

Reciprocal function: A function whose output is one divided by the output of another function, wherever that output is nonzero.
Zero: An input that makes a function's output equal to zero.
Domain: The set of input values for which a function is defined.
Vertical asymptote: A vertical line that a graph approaches near an input where the function is undefined.
Horizontal asymptote: A horizontal line that a graph approaches as inputs become very large or very negative.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation C2.1. 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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C2.1 · Graph reciprocals of linear and quadratic functions

Ontario Grade 12 Mathematics

Use zeros, signs, and selected values to sketch reciprocal graphs

What you will learn

1. Prerequisite bridge: reciprocals, zeros, and signs

2. Reciprocal of a linear function

3. Reciprocal of a quadratic function

4. A reliable sketching process

Selected points for the worked quadratic

Sketch a reciprocal quadratic

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