Q8 Connect expanded and factored quadratic forms to one graph

Ontario Grade 10 Mathematics

Quadratic Relations

Seeing why both algebraic forms describe the same parabola

You already know from Grade 9 that multiplying two binomials gives a trinomial, and that a parabola is a curved graph shaped like a U or an upside-down U. This lesson ties those two ideas together. A quadratic relation can be written in more than one algebraic form, yet every form draws the exact same parabola. Learning to move between forms — and to read key features straight from each one — is the core skill of this lesson. No new formulas are required beyond what you already know about multiplying brackets and substituting values.

What you will learn

Recognise a quadratic relation written in expanded form and in factored form.
Expand a factored quadratic and confirm both forms are equivalent.
Read the x-intercepts (zeros) and the y-intercept from the appropriate algebraic form.
Locate the axis of symmetry and vertex from coordinates already found.
Explain why two different-looking expressions can produce the same parabola.

Bridge from Grade 9: Two Forms of the Same Relation

In Grade 9 you expanded expressions like

(x+3)(x-5)

by multiplying every term in the first bracket by every term in the second bracket. The result is called the expanded form (also called standard form):

x^2 - 2x - 15

. The original product of two brackets is called the factored form. Both expressions are equal — they produce exactly the same output for every value of

x

.

A quadratic relation is any relation that can be written as

y = ax^2 + bx + c

, where

a

,

b

, and

c

are real numbers and

a \neq 0

. Its graph is always a parabola. The factored form of the same relation looks like

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

, where

r

and

s

are numbers we will identify shortly.

The key insight: if you expand

a(x-r)(x-s)

you get

ax^2 + bx + c

. Because the two forms are algebraically identical, they produce the same

y

-value for every

x

-value, so they graph as one single parabola — not two different curves.

a(x-r)(x-s) = ax^2 + bx + c

Expanded form: $y = ax^2 + bx + c$ .
Factored form: $y = a(x - r)(x - s)$ .
Both forms are equivalent — they represent one relation and one parabola.
Expanding the factored form always produces the expanded form.

Reading Key Features from Each Form

Each algebraic form makes certain features of the parabola easy to read. The factored form

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

gives you the x-intercepts immediately. An x-intercept is a point where the parabola crosses the x-axis, meaning

y = 0

. Setting each bracket equal to zero gives

x = r

and

x = s

. These values are also called the zeros or roots of the relation.

The expanded form

y = ax^2 + bx + c

gives you the y-intercept immediately. The y-intercept is where the parabola crosses the y-axis, meaning

x = 0

. Substituting

x = 0

into the expanded form gives

y = c

, so the y-intercept is always the constant term

c

.

Once you have both x-intercepts, you can find the axis of symmetry — the vertical line that cuts the parabola perfectly in half. Because the parabola is symmetric, the axis of symmetry lies exactly halfway between the two x-intercepts. Its equation is

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

.

Substituting the axis-of-symmetry value back into either form of the equation gives the y-coordinate of the vertex, which is the turning point of the parabola. The vertex sits on the axis of symmetry, so its x-coordinate equals

\frac{r+s}{2}

.

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

Zeros (x-intercepts): set $y = 0$ in the factored form to get $x = r$ and $x = s$ .
y-intercept: substitute $x = 0$ into the expanded form; the answer is $c$ .
Axis of symmetry: $x = \frac{r+s}{2}$ , halfway between the two zeros.
Vertex: substitute the axis of symmetry x-value into the equation to find $y$ .
The sign of $a$ tells you whether the parabola opens up ( $a > 0$ ) or down ( $a < 0$ ).

Expanding Factored Form to Confirm Equivalence

To verify that two forms represent the same relation, expand the factored form and check that it matches the expanded form. Use the distributive property (sometimes called FOIL) step by step: multiply the first terms, the outer terms, the inner terms, and the last terms, then collect like terms.

For example, start with

y = (x-1)(x-6)

. Expanding:

(x)(x) = x^2

,

(x)(-6) = -6x

,

(-1)(x) = -x

,

(-1)(-6) = 6

. Adding these:

x^2 - 6x - x + 6 = x^2 - 7x + 6

. So the expanded form is

y = x^2 - 7x + 6

. Both

y = (x-1)(x-6)

and

y = x^2 - 7x + 6

graph as the exact same parabola.

When

a \neq 1

, remember to distribute

a

after expanding the brackets. For example,

y = 2(x+1)(x-3)

. First expand the brackets:

(x+1)(x-3) = x^2 - 2x - 3

. Then multiply by

2

:

y = 2x^2 - 4x - 6

. You can verify the equivalence by substituting any value of

x

into both forms and checking that both give the same

y

.

(x+p)(x+q) = x^2 + (p+q)x + pq

Use the distributive property to expand bracket by bracket.
Collect like terms after expanding.
Multiply by $a$ last, after the two brackets are expanded.
Substitute a test value of $x$ into both forms to verify they match.

Connecting Both Forms to the Graph

Once you can move between forms and read key features, you can sketch or interpret a parabola without graphing technology. The process is: (1) Read the zeros

r

and

s

from the factored form. (2) Read the y-intercept

c

from the expanded form. (3) Calculate the axis of symmetry

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

. (4) Substitute that x-value into the equation to find the vertex. (5) Use the sign of

a

to decide whether the parabola opens up or down.

On a technology-generated or printed graph you can reverse this process: read the x-intercepts from the graph, write the factored form using those values, then expand to get the standard form. This shows that the graph, the factored form, and the expanded form are three views of the same mathematical object.

For a mixed-difficulty check, notice that when

a = 1

the y-intercept from the expanded form equals the product

r \times s

from the factored form (because

c = (-r)(-s) = rs

after expanding). This is a quick internal check: if

c

does not equal

rs

you have made an arithmetic error somewhere.

A single parabola can be described by one expanded form and one factored form.
Use the factored form to find zeros; use the expanded form to find the y-intercept.
The axis of symmetry and vertex require knowing the zeros first.
Reading a graph and writing algebraic forms are inverse skills — practise both directions.
When $a=1$ , the constant term in expanded form equals the product of the zeros.

What Each Form Tells You at a Glance

Feature of the Parabola	Easiest to Read From	How to Find It
Zeros (x-intercepts)	Factored form $y = a(x-r)(x-s)$	Set each bracket to zero: $x = r$ and $x = s$
y-intercept	Expanded form $y = ax^2 + bx + c$	The constant term $c$ (substitute $x = 0$ )
Axis of symmetry	After finding the zeros	$x = \frac{r+s}{2}$
Vertex	After finding axis of symmetry	Substitute axis x-value into either equation
Direction of opening	Either form — look at $a$	Up if $a > 0$ ; down if $a < 0$

Worked example

Example 1 — From Factored Form to Graph Features

A quadratic relation is given in factored form as

y = (x + 2)(x - 6)

. Without graphing technology, find: (a) the zeros, (b) the y-intercept, (c) the axis of symmetry, and (d) the vertex. Then state the expanded form.

Find the zeros
The zeros are the x-values that make $y = 0$ . Set each bracket equal to zero separately. Setting $x + 2 = 0$ gives $x = -2$ . Setting $x - 6 = 0$ gives $x = 6$ . The parabola crosses the x-axis at $(-2,\, 0)$ and $(6,\, 0)$ .
$x + 2 = 0 \Rightarrow x = -2; x - 6 = 0 \Rightarrow x = 6$
Find the y-intercept by expanding first
Expand the factored form using the distributive property so the constant term is visible. Multiply term by term: $x \cdot x = x^2$ , $x \cdot (-6) = -6x$ , $2 \cdot x = 2x$ , $2 \cdot (-6) = -12$ . Collecting like terms gives the expanded form $y = x^2 - 4x - 12$ . The constant term is $-12$ , so the y-intercept is $(0,\, -12)$ . Notice also that $(-2)(6) = -12$ , which confirms our expansion.
$y = x^2 - 6x + 2x - 12 = x^2 - 4x - 12$
Find the axis of symmetry
The axis of symmetry lies halfway between the two zeros $x = -2$ and $x = 6$ . Add the zeros and divide by 2.
$x = \frac{-2 + 6}{2} = \frac{4}{2} = 2$
Find the vertex
Substitute $x = 2$ into either form of the equation. Using the expanded form $y = x^2 - 4x - 12$ is straightforward: replace every $x$ with $2$ .
$y = (2)^2 - 4(2) - 12 = 4 - 8 - 12 = -16$
State all results
Collect the four features and the expanded form. Because $a = 1 > 0$ , the parabola opens upward. The vertex $( 2,\, -16)$ is the lowest point.

Answer: Zeros:

x = -2

and

x = 6

. y-intercept:

(0, -12)

. Axis of symmetry:

x = 2

. Vertex:

(2, -16)

. Expanded form:

y = x^2 - 4x - 12

.

Check: Substitute

x = -2

into the expanded form:

(-2)^2 - 4(-2) - 12 = 4 + 8 - 12 = 0

. ✓ Substitute

x = 6

:

(6)^2 - 4(6) - 12 = 36 - 24 - 12 = 0

. ✓ Both zeros check out, and the two forms are equivalent.

Worked example

Example 2 — Factored Form with a Leading Coefficient, Mixed Reasoning

A parabola has the equation

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

. (a) State whether the parabola opens up or down and explain why. (b) Find the zeros. (c) Find the y-intercept. (d) Find the axis of symmetry and vertex. (e) Write the expanded form and verify one zero in it.

Determine the direction of opening
The value of $a$ is $-2$ . Because $a < 0$ , the parabola opens downward. This means the vertex will be the highest point on the graph.
$a = -2 < 0$
Find the zeros
Set $y = 0$ and solve each bracket. Setting $x - 1 = 0$ gives $x = 1$ . Setting $x - 5 = 0$ gives $x = 5$ . The parabola crosses the x-axis at $(1, 0)$ and $(5, 0)$ .
$x - 1 = 0 \Rightarrow x = 1; x - 5 = 0 \Rightarrow x = 5$
Find the axis of symmetry
The axis of symmetry is halfway between $x = 1$ and $x = 5$ .
$x = \frac{1 + 5}{2} = \frac{6}{2} = 3$
Find the vertex
Substitute $x = 3$ into the factored form (either form works, but the factored form can be quicker here): replace $x$ with $3$ in $y = -2(x-1)(x-5)$ .
$y = -2(3-1)(3-5) = -2(2)(-2) = -2(-4) = 8$
Expand to get the standard form
First expand the two brackets: $(x-1)(x-5) = x^2 - 5x - x + 5 = x^2 - 6x + 5$ . Then multiply every term by $-2$ .
$y = -2(x^2 - 6x + 5) = -2x^2 + 12x - 10$
Find the y-intercept from expanded form
Substitute $x = 0$ into the expanded form. The constant term $c = -10$ , so the y-intercept is $(0, -10)$ . You can also check using the factored form: $y = -2(0-1)(0-5) = -2(-1)(-5) = -10$ . Both agree.
$y = -2(0)^2 + 12(0) - 10 = -10$

Answer: Opens downward (

a = -2 < 0

). Zeros:

x = 1

and

x = 5

. y-intercept:

(0, -10)

. Axis of symmetry:

x = 3

. Vertex:

(3, 8)

. Expanded form:

y = -2x^2 + 12x - 10

.

Check: Verify zero

x = 1

in expanded form:

-2(1)^2 + 12(1) - 10 = -2 + 12 - 10 = 0

. ✓ Verify

x = 5

:

-2(25) + 12(5) - 10 = -50 + 60 - 10 = 0

. ✓

Common mistakes and how to avoid them

Reading the zero directly from the factored form as the number that appears, e.g., writing

x = -6

instead of

x = 6

for the bracket

(x - 6)

.

Correction: Set the bracket equal to zero and solve:

x - 6 = 0

gives

x = 6

, not

-6

. The zero is the opposite sign of the number inside the bracket.

Forgetting to multiply by

a

after expanding the two brackets, e.g., leaving

y = x^2 - 6x + 5

instead of

y = -2x^2 + 12x - 10

when

a = -2

.

Correction: After expanding the brackets, distribute

a

to every term of the trinomial before writing the final expanded form.

Calculating the axis of symmetry by subtracting the zeros instead of averaging them, e.g., writing

x = 6 - (-2) = 8

instead of

x = 2

.

Correction: Add the two zeros and divide by 2:

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

. This gives the midpoint between them, not the distance.

Substituting the axis-of-symmetry value into the wrong equation and making arithmetic errors by not simplifying bracket-by-bracket.

Correction: Choose whichever form is easier, substitute carefully, and evaluate one operation at a time. Check by substituting into the other form too.

Assuming the y-intercept equals zero because the parabola has two x-intercepts.

Correction: The y-intercept is found by setting

x = 0

, not

y = 0

. Only the x-intercepts are found by setting

y = 0

.

Lesson summary

A quadratic relation can be written in expanded form $y = ax^2 + bx + c$ or in factored form $y = a(x-r)(x-s)$ ; both forms graph as the same single parabola.
The factored form makes the zeros (x-intercepts) easy to read: set each bracket to zero to get $x = r$ and $x = s$ .
The expanded form makes the y-intercept easy to read: it is the constant term $c$ , found by substituting $x = 0$ .
The axis of symmetry is the vertical line $x = \frac{r+s}{2}$ , exactly halfway between the two zeros.
Substituting the axis-of-symmetry x-value into either form of the equation gives the y-coordinate of the vertex.
Expanding the factored form using the distributive property and collecting like terms always produces the equivalent expanded form; substituting a test value into both forms confirms they are equal.

Check your understanding

Question 1

The factored form of a quadratic is

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

. What are the zeros of this relation?

$x = 4$ and $x = -2$
$x = -4$ and $x = 2$
$x = -4$ and $x = -2$
$x = 4$ and $x = 2$

Show answer and explanation

x = -4

and

x = 2

Set each bracket to zero:

x + 4 = 0

gives

x = -4

, and

x - 2 = 0

gives

x = 2

. The zero is always the opposite sign of the number inside the bracket.

Question 2

A parabola has equation

y = x^2 - 3x - 10

. What is its y-intercept?

$(0, -3)$
$(0, 3)$
$(0, -10)$
$(0, 10)$

Show answer and explanation

(0, -10)

Substitute

x = 0

into the expanded form:

y = 0 - 0 - 10 = -10

. The y-intercept is the constant term, which is

-10

.

Question 3

A parabola has zeros at

x = -1

and

x = 7

. What is the equation of its axis of symmetry?

$x = 8$
$x = 6$
$x = 4$
$x = 3$

Show answer and explanation

x = 3

The axis of symmetry is halfway between the zeros:

x = \frac{-1 + 7}{2} = \frac{6}{2} = 3

.

Question 4

Which expanded form is equivalent to

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

?

$y = -3x^2 - 6x + 24$
$y = -3x^2 + 6x - 24$
$y = -3x^2 - 6x - 24$
$y = 3x^2 + 6x - 24$

Show answer and explanation

y = -3x^2 - 6x + 24

Expand the brackets first:

(x-2)(x+4) = x^2 + 4x - 2x - 8 = x^2 + 2x - 8

. Then multiply by

-3

:

-3x^2 - 6x + 24

. The leading coefficient stays negative.

Key terms

Quadratic relation: A relation whose equation can be written as $y = ax^2 + bx + c$ with $a \neq 0$ ; its graph is always a parabola.
Expanded form (standard form): A quadratic written as $y = ax^2 + bx + c$ , where all brackets have been multiplied out and like terms collected.
Factored form: A quadratic written as $y = a(x-r)(x-s)$ , expressed as a product of two linear factors.
Zero (root): An x-value that makes $y = 0$ ; the x-coordinate of a point where the parabola crosses the x-axis.
x-intercept: A point where a graph crosses the x-axis; for a quadratic, this is the same as a zero of the relation.
y-intercept: The point where a graph crosses the y-axis, found by substituting $x = 0$ into the equation.
Axis of symmetry: A vertical line that divides the parabola into two mirror-image halves; its equation is $x = \frac{r+s}{2}$ .
Vertex: The turning point of a parabola — the highest point if the parabola opens downward, or the lowest point if it opens upward.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 10 Mathematics (MFM2P), study topic Q8. It is a study resource, not an official curriculum publication.

Official curriculum reference

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Q8 · Connect expanded and factored quadratic forms to one graph

Ontario Grade 10 Mathematics

Seeing why both algebraic forms describe the same parabola

What you will learn

Bridge from Grade 9: Two Forms of the Same Relation

Reading Key Features from Each Form

Expanding Factored Form to Confirm Equivalence

Connecting Both Forms to the Graph

What Each Form Tells You at a Glance

Example 1 — From Factored Form to Graph Features

Example 2 — Factored Form with a Leading Coefficient, Mixed Reasoning

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Question 4

Key terms

Continue through MFM2P

About this lesson