A1.8 Connect quadratic factors with x-intercepts

Ontario Grade 11 Mathematics

Mathematical Models

How a factored rule shows where a parabola meets the x-axis

A graph can show where a relation crosses or touches the x-axis. A quadratic written in factored form can show those locations directly. In this lesson, you will connect the factors to the x-values at the x-intercepts. You will also see why setting a factor equal to zero reveals the point. The focus is this connection; you do not need a new equation-solving method.

What you will learn

Identify the x-intercepts of a quadratic relation written in factored form.
Explain how each linear factor gives an x-value where the quadratic is zero.
Connect the zero values to points on the graph.

1. A short prerequisite bridge

A quadratic relation is a relation whose rule includes a squared variable, such as

y=x^2-5x+6

. Its graph is a parabola, a U-shaped or upside-down U-shaped curve. The variable

x

names horizontal positions, and

y

gives the related vertical value.

An x-intercept is a point where a graph meets the x-axis. Every point on the x-axis has a vertical coordinate of zero. So, at an x-intercept, the value of

y

is zero. For example, the point

(3,0)

is on the x-axis.

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

(x-2)(x-5)

has two factors:

(x-2)

and

(x-5)

. In factored form, the quadratic is shown as a product instead of as a sum of terms.

y=a(x-r)(x-s)

At an x-intercept, $y=0$ .
Factored form displays the expressions being multiplied.

2. Why a factor reveals an x-intercept

Consider a quadratic rule written as

y=a(x-r)(x-s)

, where

a

is a non-zero number. The factors are

(x-r)

and

(x-s)

. If

x=r

, then the first factor has value zero. Multiplying by zero makes the whole product zero, so

y=0

. This gives the x-intercept

(r,0)

.

The same reasoning applies to the other factor. If

x=s

, then

(x-s)=0

, so the product is zero and the graph has the x-intercept

(s,0)

. The factor tells you which x-value makes the quadratic zero. The intercept is the full point, so include a zero as its y-coordinate.

A negative sign inside a factor matters. The factor

(x+4)

becomes zero when

x=-4

, not when

x=4

. Think of the value that makes the whole factor equal zero. A non-zero number in front of the factors changes the graph's vertical shape, but it does not change which factor values make the product zero.

x-r=0\;\Rightarrow\;x=r

For a factor $(x-r)$ , the matching x-intercept is $(r,0)$ .
Use the value that makes the factor zero; do not copy its sign without checking.

3. Read factors, then connect to the graph

A useful way to read factored form is to consider each factor separately. Set that factor to zero and find the x-value. Then write the intercept as an ordered pair with zero for its y-coordinate. An ordered pair lists the horizontal coordinate first and the vertical coordinate second.

For example, the factors

(x-1)

and

(x+3)

point to x-values of

1

and

-3

. The corresponding points are

(1,0)

and

(-3,0)

. These are where the graph meets the x-axis. This does not tell you every feature of the graph, but it identifies its x-intercepts.

You can check the connection by substituting an intercept's x-value into the factored rule. If either factor becomes zero, the whole product must be zero. This is why the factor-to-intercept connection works, rather than just a pattern to memorize.

y=0\;\text{at an x-intercept}

Set each factor to zero to find its x-value.
Write each x-intercept as $(x,0)$ .
Substitution checks that the quadratic output is zero.

4. Apply the connection carefully

When a quadratic is already factored, you do not need to expand it to find its x-intercepts. Read each factor, find the value that makes it zero, and write the matching point. Keeping the expression factored makes the information easy to see.

If the same factor appears twice, it still becomes zero at the same x-value. That means the graph has an x-intercept at that value; it does not create a second different x-value. For this lesson, focus on identifying the x-value and writing its intercept point.

In the examples, notice the distinction between an x-value and an intercept. A factor leads to a number such as

-2

. The graph location is the point

(-2,0)

. Including both coordinates clearly states where the graph meets the axis.

A repeated factor gives the same x-value, not a new intercept location.
State intercepts as points, not only as numbers.

Reading a factor

Factor	Value that makes it zero	x-intercept
$(x-6)$	$x=6$	$(6,0)$
$(x+3)$	$x=-3$	$(-3,0)$

Worked example

Two factors with different signs

Find the x-intercepts of

y=(x-4)(x+2)

.

Use the first factor
The graph meets the x-axis when $y=0$ . The first factor is zero when $x=4$ , which makes the entire product zero.
$x-4=0\;\Rightarrow\;x=4$
Use the second factor
The second factor is zero when $x=-2$ . The plus sign in $(x+2)$ means the value that cancels the 2 is negative.
$x+2=0\;\Rightarrow\;x=-2$
Write the points
Each x-value is an intercept with y-coordinate zero. The intercepts are the points $(4,0)$ and $(-2,0)$ .
$(4,0),\;(-2,0)$

Answer: The x-intercepts are

(4,0)

and

(-2,0)

.

Check: At

x=4

, the factor

(x-4)

is zero. At

x=-2

, the factor

(x+2)

is zero. In each case,

y=0

.

Worked example

A coefficient does not change the intercepts

Find the x-intercepts of

y=3(x+1)(x-5)

.

Find the first factor value
The factor $(x+1)$ is zero when $x=-1$ . Then the product is zero, even though the rule also has a factor of 3.
$x+1=0\;\Rightarrow\;x=-1$
Find the second factor value
The factor $(x-5)$ is zero when $x=5$ . This also makes the product zero.
$x-5=0\;\Rightarrow\;x=5$
Connect values to points
The x-intercepts are the points with those x-values and a y-coordinate of zero. The coefficient 3 does not change which factor values make the product zero.
$(-1,0),\;(5,0)$

Answer: The x-intercepts are

(-1,0)

and

(5,0)

.

Check: Substituting either x-value makes one factor zero, so the full product, including its coefficient, equals zero.

Common mistakes and how to avoid them

Reading

(x+3)

as giving the x-value

3

.

Correction: Find the value that makes the factor zero. For

(x+3)

, that value is

x=-3

.

Writing only the x-values as the intercepts.

Correction: An x-intercept is a point on the graph. Write the y-coordinate as zero, such as

(6,0)

.

Thinking a non-zero coefficient in front changes the x-intercepts.

Correction: The coefficient changes the product's size, but the product is still zero when either factor is zero.

Lesson summary

An x-intercept is a point where the quadratic graph meets the x-axis, so its y-coordinate is zero.
In factored form, set each factor equal to zero to find the corresponding x-value.
Turn each x-value into an intercept point by pairing it with zero.
Check the signs by making each factor equal zero.

Check your understanding

Question 1

What are the x-intercepts of

y=(x-7)(x+2)

?

$(7,0)$ and $(-2,0)$
$(-7,0)$ and $(2,0)$
$(7,2)$ and $(0,0)$
correctIndex":0,"explanation":"The first factor is zero at $x=7$ , and the second is zero at $x=-2$ . The intercept points have y-coordinate zero."

Show answer and explanation

(7,0)

and

(-2,0)

The first factor is zero at

x=7

, and the second is zero at

x=-2

. The intercept points have y-coordinate zero.

Question 2

For

y=2(x+4)(x-1)

, which x-value comes from the factor

(x+4)

?

$x=4$
$x=-4$
$x=1$
correctIndex":1,"explanation":"The factor $(x+4)$ equals zero when $x=-4$ , since $-4+4=0$ ."

Show answer and explanation

x=-4

The factor

(x+4)

equals zero when

x=-4

, since

-4+4=0

.

Key terms

Quadratic relation: A relation with a squared variable whose graph is a parabola.
Factor: One of the expressions multiplied together in a product.
Factored form: A way to write a quadratic as a product of factors.
x-intercept: A point where a graph meets the x-axis; its y-coordinate is zero.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation A1.8. It is a study resource, not an official curriculum publication.

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A1.8 · Connect quadratic factors with x-intercepts