C3.3 Connect real roots with x-intercepts | Ontario Grade 12 Mathematics

Ontario Grade 12 Mathematics

Polynomial and Rational Functions

Connecting solutions of an equation to points on a graph

A graph shows where a function’s output is zero. An equation identifies the input values that make the output zero. These are two ways to describe the same locations. In this lesson, you will connect real roots with the points where a function’s graph meets the

x

-axis.

What you will learn

Explain what a real root of a function is.
Identify the x-intercepts of a function’s graph.
Connect solutions of $f(x)=0$ with x-intercepts.
Check a proposed root by evaluating the function.

1. Prerequisite bridge: inputs, outputs, and axes

A function pairs an input with an output. We often name the input

x

and write the output as

f(x)

. For example, if

f(2)=0

, the input is

2

and the output is

0

.

On a coordinate grid, the horizontal axis is the

x

-axis. Every point on it has a vertical coordinate of zero. A point on this axis has the form

(x,0)

. The vertical axis is the

y

-axis. Every point on it has a horizontal coordinate of zero.

The graph of

y=f(x)

contains points

(x,f(x))

. Each point records an input and its corresponding output. If the output is zero, the point is

(x,0)

, so it lies on the

x

-axis.

y=f(x)

The first coordinate of a graph point is the input.
The second coordinate is the output.
Every point on the $x$ -axis has a vertical coordinate of zero.

2. What a real root means

A root of a function is an input value that makes the function’s output equal to zero. A real root is a root that is a real number, such as

-3

,

0

, or

2.5

.

To find roots algebraically, set the function equal to zero and solve for

x

. Each real solution gives a graph point

(x,0)

, because the output at that input is zero. This point is an

x

-intercept: a point where the graph meets the

x

-axis.

For example, if

f(4)=0

, then

4

is a real root and

(4,0)

is an

x

-intercept of the graph of

y=f(x)

. The root is the input value

4

. The intercept is the point

(4,0)

. They are connected, but they are not the same kind of object.

A graph may meet the

x

-axis at more than one point, or not meet it at all. Solving

f(x)=0

identifies the real input values that correspond to the graph’s

x

-intercepts. A graph can help you see these locations, while solving the equation can give exact input values.

For any real number

r

in the function’s domain,

f(r)=0

means that the graph of

y=f(x)

contains the point

(r,0)

. In the other direction, if

(r,0)

is on the graph, then

f(r)=0

.

A real root is a real input that makes the function equal to zero.
An $x$ -intercept is a graph point with vertical coordinate zero.
A real root $r$ corresponds to the $x$ -intercept $(r,0)$ .
To find roots, solve $f(x)=0$ .

3. Three representations of the same connection

Consider an input for which a function’s output is zero. The equation, function value, and graph point express the same fact in different ways.

In equation form, solve

f(x)=0

. In function-value form, state that

f(r)=0

for a particular real number

r

. In graph form, identify

(r,0)

on the

x

-axis. Moving between these forms helps keep a root separate from an intercept.

Reading a location from a graph is often an estimate. If a graph appears to meet the axis near

x=2

, check the equation before claiming the exact root is

2

. A drawing may not show every coordinate precisely.

In symbols, the connection is f(r)=0 ext{ if and only if } (r,0) ext{ lies on the graph of }y=f(x). Here, “if and only if” means that either statement implies the other.

Equation: solve $f(x)=0$ .
Function value: check whether $f(r)=0$ .
Graph: locate the corresponding point $(r,0)$ .
The root is the intercept’s $x$ -coordinate; the intercept is the full ordered pair.

4. Applying the connection carefully

When asked for roots, report input values, not points. When asked for

x

-intercepts, report ordered pairs. For example, the root

-1

corresponds to the intercept

(-1,0)

.

To check a proposed root, substitute it into the function. If the result is zero, it is a root and the corresponding graph point lies on the

x

-axis. If the result is not zero, it is not a root, and the corresponding point is not an

x

-intercept.

A graph and an equation can be used together. The graph shows where the output is zero. The equation identifies the inputs that make it zero. The central connection is that real roots correspond to the graph’s

x

-intercepts.

f(r)=0\Rightarrow(r,0)

Report roots as real numbers.
Report intercepts as points with second coordinate zero.
Verify a proposed root by evaluating the function.

One fact in three forms

Representation	What it says	Example from the worked function
Equation	Set the output to zero.	$x^2-x-6=0$
Root	An input that makes the output zero.	$x=-2$ or $x=3$
x-intercept	The graph point with that input and output zero.	$(-2,0)$ or $(3,0)$

Worked example

From an equation to x-intercepts

For

f(x)=x^2-x-6

, find the real roots and state the corresponding

x

-intercepts of the graph

y=f(x)

.

Set the output to zero
A root is an input that makes the function’s output zero. Begin by setting $f(x)$ equal to zero.
$x^2-x-6=0$
Factor the expression
Find two numbers whose product is $-6$ and whose sum is $-1$ . The numbers $-3$ and $2$ work, so the expression factors as shown.
$(x-3)(x+2)=0$
Find the input values
For a product to equal zero, at least one factor must equal zero. Solving each factor gives the real roots $3$ and $-2$ . x=3 or x=-2
Connect roots to graph points
At each root, the function’s output is zero. Therefore, each root gives a point on the $x$ -axis. The root $3$ gives $(3,0)$ , and the root $-2$ gives $(-2,0)$ . (3,0) and (-2,0)

Answer: The real roots are

-2

and

3

. The

x

-intercepts are

(-2,0)

and

(3,0)

.

Check: Substitution confirms both values:

f(-2)=4+2-6=0

and

f(3)=9-3-6=0

. Each corresponding graph point therefore has vertical coordinate zero.

Common mistakes and how to avoid them

Calling the point

(-2,0)

the root.

Correction: The root is the input value

-2

. The corresponding

x

-intercept is the point

(-2,0)

.

Giving an intercept as only a number, such as

3

.

Correction: An intercept is a point, so write

(3,0)

. The number

3

is the root.

Using a point with a nonzero vertical coordinate as an

x

-intercept.

Correction: An

x

-intercept must lie on the

x

-axis, so its vertical coordinate must be zero.

Assuming a possible root is correct without checking it.

Correction: Substitute the input into the function. It is a root only if the output is zero.

Lesson summary

A real root is an input value that makes a function’s output zero.
To find roots, solve $f(x)=0$ .
If $r$ is a real root, then $(r,0)$ is an $x$ -intercept of $y=f(x)$ .
Roots are numbers; $x$ -intercepts are ordered pairs.

Check your understanding

Question 1

If

f(5)=0

, which statement is correct?

The root is the point $(5,0)$ .
The root is $5$ , and the corresponding $x$ -intercept is $(5,0)$ .
The $x$ -intercept is $(0,5)$ .
The function has no real root.

Show answer and explanation

The root is

5

, and the corresponding

x

-intercept is

(5,0)

.

The input

5

makes the output zero, so

5

is the root and

(5,0)

is the point on the

x

-axis.

Question 2

A graph of

y=g(x)

has an

x

-intercept at

(-4,0)

. What does this tell you?

$g(0)=-4$
$g(-4)=0$
$g(4)=0$
$g(-4)=4$

Show answer and explanation

g(-4)=0

The point’s input is

-4

and its output is zero, so

g(-4)=0

. Thus,

-4

is a real root.

Question 3

Which equation should you solve to find the real roots of

h(x)

?

$h(x)=1$
$x=0$
$h(x)=0$
$h(0)=x$

Show answer and explanation

h(x)=0

Roots are inputs for which the function’s output is zero, so solve

h(x)=0

.

Key terms

Function: A rule that assigns an output to each allowed input.
Real root: A real input value that makes a function’s output equal zero.
x-intercept: A point where a graph meets the horizontal $x$ -axis. Its vertical coordinate is zero.
Ordered pair: A pair of coordinates written $(x,y)$ , with the input first and output second.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation C3.3. It is a study resource, not an official curriculum publication.

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C3.3 · Connect real roots with x-intercepts

Ontario Grade 12 Mathematics

Connecting solutions of an equation to points on a graph

What you will learn

1. Prerequisite bridge: inputs, outputs, and axes

2. What a real root means

3. Three representations of the same connection

4. Applying the connection carefully

One fact in three forms

From an equation to x-intercepts

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