A2.9 Sketch a quadratic from factored form | Ontario Grade 11 Mathematics

Ontario Grade 11 Mathematics

Quadratic Functions

Use zeros, symmetry, the vertex, and opening direction to guide your sketch.

Factored form shows useful information about a quadratic graph. Each factor helps you find a point where the graph meets the horizontal axis. The two points help locate the axis of symmetry and the vertex. The number multiplying the factors tells you whether the parabola opens upward or downward. Together, these features are enough to make a clear sketch.

What you will learn

Find the zeros of a quadratic written in factored form.
Use the zeros to locate the axis of symmetry.
Find the vertex by substituting into the equation.
Sketch the parabola using its key features.

Prerequisite bridge: factors and coordinates

A coordinate is an ordered pair that names a point on a graph. The first number is the horizontal position, or

x

-coordinate. The second is the vertical position, or

y

-coordinate. A point on the horizontal axis has a

y

-coordinate of zero.

A zero is an input value that makes the output equal to zero. To find a zero from a factor, set that factor equal to zero and solve. For example,

x-4=0

gives

x=4

, while

x+2=0

gives

x=-2

.

Each zero gives a point where the graph meets the horizontal axis. A zero of

4

corresponds to the point

(4,0)

.

y=0

Set each factor equal to zero and solve.
A zero $r$ gives the point $(r,0)$ .

Read the key features from factored form

A quadratic in factored form can be written as

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

. The values

r

and

s

are the zeros. The number

a

is the leading coefficient: it is the number multiplying the factors.

The graph of a quadratic is a parabola. If

a

is positive, the parabola opens upward. If

a

is negative, it opens downward.

The axis of symmetry is a vertical line that divides the parabola into matching left and right sides. It lies halfway between the zeros. The vertex is the turning point of the parabola and lies on this axis.

To find the vertex, first find the axis value. This gives the vertex’s

x

-coordinate. Substitute that value into the equation to find the vertex’s

y

-coordinate. For an upward-opening parabola, the vertex is its lowest point. For a downward-opening parabola, it is its highest point.

x=\frac{r+s}{2}

The zeros give the horizontal-axis intercepts.
The axis of symmetry is halfway between the zeros.
The vertex lies on the axis of symmetry.
The sign of the leading coefficient gives the opening direction.

From features to a sketch

Start by plotting the two zeros on the horizontal axis. Then locate the vertical axis of symmetry halfway between them. Mark the vertex after you have calculated its height.

You can also find the vertical-axis intercept, the point where the graph meets the vertical axis. Every point on that axis has an

x

-coordinate of zero, so substitute

x=0

into the equation. This extra point can help shape your sketch.

Draw a smooth curve through the points. Make the two sides match around the axis of symmetry. Your sketch should show the zeros, the vertex, the axis, and the opening direction.

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

Plot the zeros and vertex as coordinates.
Use the axis of symmetry to keep the two sides balanced.
Find the vertical-axis intercept, if useful, by substituting $x=0$ .

A reliable sketching routine

Read each factor and set it equal to zero. Plot both zeros. Find their midpoint to get the axis of symmetry.

Substitute the axis value into the equation to calculate the vertex. Check the sign of the leading coefficient to decide whether the parabola opens upward or downward.

Before completing the sketch, check that the curve passes through both zeros and the vertex. If you found the vertical-axis intercept, include it too. Draw a smooth curve with matching sides.

Find and plot the zeros first.
Calculate the vertex height rather than guessing it.
Use the opening direction and symmetry to complete the sketch.

Key features for the example

Feature	Value	What to mark
Zeros	$x=-1$ and $x=5$	$(-1,0)$ and $(5,0)$
Axis of symmetry	$x=2$	Vertical line through $x=2$
Vertex	$(2,18)$	Turning point
Opening	Downward	Curve opens downward from the vertex
Vertical-axis intercept	$(0,10)$	Point where $x=0$

Worked example

Sketch a quadratic with two zeros

Sketch

y=-2(x+1)(x-5)

. Show the zeros, axis of symmetry, vertex, opening direction, and vertical-axis intercept.

Find the zeros
Set each factor equal to zero. Solving $x+1=0$ gives $x=-1$ , and solving $x-5=0$ gives $x=5$ . The corresponding points are $(-1,0)$ and $(5,0)$ .
$x=-1,\quad x=5$
Find the axis
The axis is halfway between the zeros. Their midpoint gives the vertex’s horizontal coordinate, so the axis is the vertical line through that value.
$x=\frac{-1+5}{2}=2$
Find the vertex
Substitute $x=2$ into the equation. The output is $18$ , so the vertex is $(2,18)$ . Since the parabola opens downward, the vertex is its highest point.
$y=-2(2+1)(2-5)=18$
Find the opening and vertical-axis intercept
The leading coefficient is negative, so the parabola opens downward. To find the vertical-axis intercept, substitute $x=0$ . The output is $10$ , giving the point $(0,10)$ .
$y=-2(0+1)(0-5)=10$
Complete the sketch
Plot the two zeros, the vertex, and the vertical-axis intercept. Draw a smooth downward-opening curve through them. The two sides should match around the axis $x=2$ .

Answer: The zeros are

(-1,0)

and

(5,0)

. The axis of symmetry is

x=2

, the vertex is

(2,18)

, and the vertical-axis intercept is

(0,10)

. The parabola opens downward.

Check: The midpoint of

-1

and

5

is

2

, which matches the vertex’s horizontal coordinate. Substitution gives a vertex height of

18

and a vertical-axis intercept height of

10

. The negative leading coefficient agrees with the downward opening.

Common mistakes and how to avoid them

Reading the zero from a factor without solving, such as saying

x+1

gives

x=1

.

Correction: Set the factor equal to zero and solve. For example,

x+1=0

gives

x=-1

.

Using one zero as the axis of symmetry.

Correction: Find the midpoint of the two zero coordinates. The axis is the vertical line through that midpoint.

Treating the axis value as the whole vertex.

Correction: The axis gives the vertex’s

x

-coordinate only. Substitute it into the equation to find the

y

-coordinate.

Assuming a parabola opens upward because it has two zeros.

Correction: Check the leading coefficient. A negative leading coefficient means the parabola opens downward.

Drawing sides that do not match around the axis.

Correction: Use the axis of symmetry as a guide. The left and right sides should match.

Lesson summary

Set each factor equal to zero to find the zeros and horizontal-axis intercepts.
Find the axis of symmetry halfway between the zeros.
Substitute the axis value into the equation to find the vertex.
Use the sign of the leading coefficient to decide the opening direction.
Plot the key points and draw a smooth, symmetric parabola.

Check your understanding

Question 1

For

y=(x-2)(x-8)

, which statement gives the zeros and axis of symmetry?

Zeros $2$ and $8$ ; axis $x=5$
Zeros $-2$ and $-8$ ; axis $x=-5$
Zeros $2$ and $8$ ; axis $x=8$
Zeros $-2$ and $8$ ; axis $x=3$

Show answer and explanation

Zeros

2

and

8

; axis

x=5

The factors give zeros

x=2

and

x=8

. Their midpoint is

5

, so the axis is

x=5

.

Question 2

For

y=-3(x+2)(x-4)

, which way does the parabola open?

Upward, because there are two factors
Downward, because the leading coefficient is negative
Upward, because the zeros have different signs
Downward, because the axis of symmetry is vertical

Show answer and explanation

Downward, because the leading coefficient is negative

The leading coefficient is

-3

. A negative leading coefficient means the parabola opens downward.

Key terms

Quadratic: A function whose graph is a parabola.
Factored form: A way to write a quadratic as a number multiplied by two factors.
Zero: An input value that makes a function’s output equal to zero.
Axis of symmetry: A vertical line that divides a parabola into matching left and right sides.
Vertex: The turning point of a parabola.
Leading coefficient: The number multiplying the factors in a quadratic written in factored form.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation A2.9. It is a study resource, not an official curriculum publication.

Official curriculum reference

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A2.9 · Sketch a quadratic from factored form

Ontario Grade 11 Mathematics

Use zeros, symmetry, the vertex, and opening direction to guide your sketch.

What you will learn

Prerequisite bridge: factors and coordinates

Read the key features from factored form

From features to a sketch

A reliable sketching routine

Key features for the example

Sketch a quadratic with two zeros

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Key terms

Continue through MCF3M

About this lesson