3. Factored Form

Learning outcomes

I can identify quadratic equations written in factored form.
I can determine the x-intercepts from factored form.
I can identify the axis of symmetry using the roots.
I can sketch graphs from factored form.
I can compare factored form with other forms.

Factored Form

What Is Factored Form?

A quadratic function can be written in factored form:

y = a(x − r₁)(x − r₂)

where:

a affects the direction and vertical stretch or compression of the parabola
r₁ is one root or x-intercept
r₂ is the other root or x-intercept

Factored form is especially useful because the x-intercepts can usually be identified immediately.

For example:

y = (x − 2)(x − 6)

has x-intercepts:

x = 2 and x = 6

or, as coordinate points:

(2, 0) and (6, 0)

Roots, Zeros and x-Intercepts

The terms roots, zeros, and x-intercepts are closely related.

A root or zero is an x-value that makes:

y = 0

An x-intercept is the corresponding point where the graph crosses or touches the x-axis.

For example:

y = (x − 3)(x + 5)

The roots are:

x = 3

and:

x = −5

The x-intercepts are:

(3, 0) and (−5, 0)

Finding the x-Intercepts

To find an x-intercept, set:

y = 0

Consider:

y = (x − 2)(x − 7)

Set y = 0:

0 = (x − 2)(x − 7)

For the product to equal zero, at least one factor must equal zero.

So:

x − 2 = 0

x = 2

or:

x − 7 = 0

x = 7

Therefore, the x-intercepts are:

(2, 0) and (7, 0)

The Zero Product Property

This method relies on the Zero Product Property.

If:

AB = 0

then:

A = 0 or B = 0

For a quadratic:

(x − r₁)(x − r₂) = 0

we can therefore write:

x − r₁ = 0

or:

x − r₂ = 0

giving:

x = r₁ or x = r₂

This is why factored form makes the roots so easy to identify.

Watch the Signs

As with vertex form, the signs can cause mistakes.

For:

(x − 4)

the root is:

x = 4

But for:

(x + 4)

the root is:

x = −4

because:

x + 4 = 0

x = −4

The sign of the root is therefore the opposite of the sign shown inside the factor.

Examples of Finding Roots

Factored Form	Roots	x-Intercepts
y = (x − 2)(x − 5)	2, 5	(2, 0), (5, 0)
y = (x + 3)(x − 4)	−3, 4	(−3, 0), (4, 0)
y = 2(x − 1)(x − 6)	1, 6	(1, 0), (6, 0)
y = −3(x + 5)(x + 2)	−5, −2	(−5, 0), (−2, 0)
y = ½x(x − 8)	0, 8	(0, 0), (8, 0)

Notice that a does not change the roots.

Axis of Symmetry from the Roots

A parabola is symmetrical.

If its roots are:

r₁ and r₂

the axis of symmetry lies exactly halfway between them.

Therefore:

x = (r₁ + r₂)/2

This is simply the average of the two roots.

Example: Finding the Axis of Symmetry

Consider:

y = (x − 2)(x − 8)

The roots are:

r₁ = 2

r₂ = 8

Find their midpoint:

x = (2 + 8)/2

x = 10/2

x = 5

Therefore, the axis of symmetry is:

x = 5

Why the Axis Is Halfway Between the Roots

The two sides of a parabola are mirror images.

If the graph crosses the x-axis at:

(2, 0)

and:

(8, 0)

the axis of symmetry must be exactly halfway between those points.

The distance from 2 to 5 is 3.

The distance from 5 to 8 is also 3.

So:

x = 5

is the line of symmetry.

Finding the Vertex from Factored Form

Factored form does not show the vertex directly, but we can find it easily.

Consider:

y = (x − 2)(x − 8)

We already know the axis of symmetry:

x = 5

The vertex lies on this line.

Substitute x = 5:

y = (5 − 2)(5 − 8)

y = (3)(−3)

y = −9

Therefore:

vertex = (5, −9)

A Reliable Vertex Method

When given factored form:

y = a(x − r₁)(x − r₂)

Find the roots

r₁ and r₂

Find the axis of symmetry

x = (r₁ + r₂)/2

Substitute this x-value into the function

The resulting y-value gives the vertex.

This gives:

vertex = (x, y)

The Role of a

The value a has the same basic role in all forms of a quadratic.

If:

a > 0

the parabola opens upward.

If:

a < 0

the parabola opens downward.

Also:

|a| > 1 → vertically stretched, appearing narrower
|a| = 1 → basic width
0 < |a| < 1 → vertically compressed, appearing wider

Example with a Negative Value of a

Consider:

y = −2(x − 1)(x − 5)

Roots:

x = 1 and x = 5

Axis of symmetry:

x = (1 + 5)/2

x = 3

Find the vertex:

y = −2(3 − 1)(3 − 5)

y = −2(2)(−2)

y = 8

Therefore:

vertex = (3, 8)

Since:

a = −2

the parabola opens downward.

The vertex is therefore a maximum.

Finding the y-Intercept

Factored form also allows us to find the y-intercept.

At the y-intercept:

x = 0

Consider:

y = (x − 2)(x − 6)

Substitute x = 0:

y = (0 − 2)(0 − 6)

y = (−2)(−6)

y = 12

Therefore, the y-intercept is:

(0, 12)

The y-Intercept in General

For:

y = a(x − r₁)(x − r₂)

set x = 0:

y = a(−r₁)(−r₂)

Therefore:

y = ar₁r₂

So the y-intercept is:

(0, ar₁r₂)

This is useful when sketching.

Sketching from Factored Form

Factored form gives us several important features needed for a sketch.

Consider:

y = (x + 1)(x − 5)

Find the roots

x = −1

x = 5

So the x-intercepts are:

(−1, 0) and (5, 0)

Find the axis of symmetry

x = (−1 + 5)/2

x = 2

Find the vertex

Substitute x = 2:

y = (2 + 1)(2 − 5)

y = (3)(−3)

y = −9

Vertex:

(2, −9)

Determine direction

a = 1

Therefore, the parabola opens upward.

Find the y-intercept

Set x = 0:

y = (1)(−5)

y = −5

So:

y-intercept = (0, −5)

We now have enough information to sketch the graph accurately.

A Quick Sketching Strategy

When sketching from factored form, find these features:

x-intercepts
axis of symmetry
vertex
direction of opening
y-intercept

Then draw a smooth symmetrical parabola through the important points.

Repeated Roots

Sometimes both factors are identical.

For example:

y = (x − 3)²

This can also be written:

y = (x − 3)(x − 3)

Both roots are:

x = 3

This is called a repeated root or double root.

The parabola touches the x-axis at:

(3, 0)

but does not cross it.

The x-intercept is also the vertex.

Crossing vs Touching the x-Axis

If a quadratic has two different real roots, the graph usually crosses the x-axis twice.

For example:

y = (x − 2)(x − 6)

crosses at:

x = 2 and x = 6

If the quadratic has a repeated root, the graph touches the x-axis once.

For example:

y = (x − 4)²

touches at:

x = 4

This difference is important when interpreting quadratic graphs.

Factored Form and Standard Form

Standard form is:

y = ax² + bx + c

Factored form is:

y = a(x − r₁)(x − r₂)

The same quadratic can often be written in either form.

For example:

y = (x − 2)(x − 5)

Expand:

y = x² − 5x − 2x + 10

Therefore:

y = x² − 7x + 10

So:

y = (x − 2)(x − 5)

and:

y = x² − 7x + 10

represent the same parabola.

What Each Form Reveals

From:

y = (x − 2)(x − 5)

we can immediately identify:

roots = 2 and 5

From:

y = x² − 7x + 10

we can immediately identify:

y-intercept = 10

Different forms make different features easier to see.

Factored Form and Vertex Form

Vertex form is:

y = a(x − h)² + k

Factored form is:

y = a(x − r₁)(x − r₂)

Vertex form makes it easy to identify:

vertex
axis of symmetry
maximum or minimum
transformations

Factored form makes it easy to identify:

roots
x-intercepts
axis of symmetry from the roots

Comparing the Three Main Forms

Form	General Equation	Information Seen Most Easily
Standard	y = ax² + bx + c	y-intercept
Vertex	y = a(x − h)² + k	vertex and axis of symmetry
Factored	y = a(x − r₁)(x − r₂)	roots and x-intercepts

The coefficient a can be identified directly in all three forms.

One Quadratic in Three Forms

Consider the quadratic:

y = x² − 6x + 5

Standard Form

y = x² − 6x + 5

The y-intercept is immediately visible:

(0, 5)

Factored Form

Factor:

y = (x − 1)(x − 5)

The roots are immediately visible:

x = 1 and x = 5

Vertex Form

The same function can be written:

y = (x − 3)² − 4

The vertex is immediately visible:

(3, −4)

All three equations represent exactly the same parabola:

y = x² − 6x + 5

y = (x − 1)(x − 5)

y = (x − 3)² − 4

The graphs overlap because the equations are equivalent.

Choosing the Most Useful Form

Use factored form when you want to find:

roots
zeros
x-intercepts
where a quadratic changes sign

Use vertex form when you want to find:

vertex
maximum or minimum
axis of symmetry
transformations

Use standard form when you want to find:

y-intercept
coefficients
a convenient form for algebraic operations

Choosing the right form can make a problem much easier.

Interpreting Roots in Real Situations

The roots of a quadratic often represent important events.

Suppose the height of an object is modeled by:

h = −5(t − 1)(t − 7)

The roots are:

t = 1 and t = 7

At these times:

h = 0

Depending on the situation, these might represent times when the object is at ground level.

Factored form makes these times immediately visible.

Example: Projectile Motion

Suppose:

h = −4(t − 2)(t − 8)

The height is zero when:

t − 2 = 0

or:

t − 8 = 0

Therefore:

t = 2 and t = 8

The axis of symmetry is:

x = (2 + 8)/2

t = 5

This means the maximum height occurs at:

t = 5

Find the maximum height:

h = −4(5 − 2)(5 − 8)

h = −4(3)(−3)

h = 36

Therefore, the vertex is:

(5, 36)

and the maximum height is:

36 units

Positive and Negative Regions

Factored form can also help determine where a quadratic is positive or negative.

Consider:

y = (x − 2)(x − 6)

The roots divide the x-axis into three regions:

x < 2

2 < x < 6

x > 6

Since the parabola opens upward:

y > 0 when x < 2
y < 0 when 2 < x < 6
y > 0 when x > 6

This becomes useful later when solving quadratic inequalities.

Worked Example

Analyze:

y = −(x + 2)(x − 4)

Roots

x = −2

x = 4

x-intercepts

(−2, 0) and (4, 0)

Axis of symmetry

x = (−2 + 4)/2

x = 1

Vertex

Substitute x = 1:

y = −(1 + 2)(1 − 4)

y = −(3)(−3)

y = 9

Vertex:

(1, 9)

Direction

a = −1

Therefore, the parabola opens downward.

y-intercept

Set x = 0:

y = −(2)(−4)

y = 8

So:

y-intercept = (0, 8)

The graph therefore has:

roots at −2 and 4
axis x = 1
maximum at (1, 9)
y-intercept (0, 8)

Common Mistakes

Mistake: The root of (x − 5) is −5.
Set the factor equal to zero:

x − 5 = 0

so x = 5.

Mistake: The root of (x + 3) is 3.
The root is x = −3.

Mistake: The axis of symmetry is the sum of the roots.
It is their average:

x = (r₁ + r₂)/2

Mistake: The coefficient a changes the roots.
A nonzero value of a changes the shape and direction but not the roots shown by the factors.

Mistake: The roots are points such as x = 2.
Strictly, x = 2 is a root. The corresponding x-intercept is the point (2, 0).

Mistake: Every parabola crosses the x-axis twice.
A parabola can have two distinct real roots, one repeated real root, or no real roots.

Mistake: Factored form and vertex form describe different types of quadratic functions.
They are simply different representations of quadratic functions.

Key Terms

Factored form — quadratic form y = a(x − r₁)(x − r₂).

Factor — expression multiplied by another expression to form a product.

Root — x-value that makes the function equal to zero.

Zero — another term for a root of a function.

x-intercept — point where a graph meets the x-axis.

Zero Product Property — if a product equals zero, at least one factor must equal zero.

Axis of symmetry — vertical line dividing a parabola into two mirror-image halves.

Vertex — maximum or minimum point of a parabola.

Repeated root — root that occurs twice in a quadratic.

Double root — another term for a repeated root.

Standard form — y = ax² + bx + c.

Vertex form — y = a(x − h)² + k.

Key Takeaways

Factored form is y = a(x − r₁)(x − r₂).
The values r₁ and r₂ give the roots of the quadratic.
The corresponding x-intercepts are (r₁, 0) and (r₂, 0).
To find a root, set each factor equal to zero.
The Zero Product Property explains why this method works.
The sign inside a factor is opposite to the sign of its root.
The axis of symmetry lies halfway between the two roots.
The axis can be found using x = (r₁ + r₂)/2.
The vertex lies on the axis of symmetry.
Substitute the axis value into the function to find the y-coordinate of the vertex.
If a > 0, the parabola opens upward.
If a < 0, the parabola opens downward.
The y-intercept can be found by setting x = 0.
A repeated root occurs when both factors give the same root.
At a repeated root, the parabola touches the x-axis rather than crossing it.
Factored form makes roots easiest to identify.
Vertex form makes the vertex easiest to identify.
Standard form makes the y-intercept easiest to identify.
All three forms can represent exactly the same quadratic function.

Check Your Understanding

Identify the roots of y = (x − 3)(x − 7).
Identify the x-intercepts of y = (x + 4)(x − 2).
Find the axis of symmetry of y = (x − 2)(x − 10).
Find the vertex of y = (x − 2)(x − 6).
Determine the roots, axis of symmetry and vertex of y = −(x + 1)(x − 5).
Does y = 3(x − 4)(x − 8) open upward or downward?
Find the y-intercept of y = (x − 3)(x − 5).
What is special about the graph of y = (x − 6)²?
Sketch y = (x + 2)(x − 4) using its intercepts, axis and vertex.
Expand y = (x − 2)(x − 7) into standard form.
Write y = x² − 8x + 15 in factored form.
Compare what can be identified immediately from factored form, vertex form and standard form.
A projectile is modeled by h = −5(t − 1)(t − 9). At what times is its height zero?
For the same projectile, at what time does it reach its maximum height, and what is that maximum height?