Convert vertex form to standard form

Ontario Grade 11 Mathematics

Quadratic Functions

Expand the squared binomial, then collect like terms.

A quadratic expression can be written in different forms. Vertex form shows the vertex of its graph directly. Standard form makes the squared term, linear term, and constant easy to identify. In this lesson, you will convert vertex form to standard form by expanding and simplifying. The expressions describe the same quadratic; only their written forms change.

What you will learn

Recognize vertex form and standard form for a quadratic expression.
Expand a squared binomial correctly.
Use the distributive property and collect like terms to convert vertex form to standard form.
Check that the converted expression is equivalent to the original.

1. Prerequisite bridge: expand and collect

Before converting, review two skills. The distributive property means multiplying a factor by every term inside parentheses. For example,

3(x+2)=3x+6

. Like terms have the same variable part. You can combine

4x

and

-x

to get

3x

, but you cannot combine

x^2

and

x

.

A squared binomial is two terms in parentheses raised to the power of two. Squaring the parentheses means multiplying the entire binomial by itself. It does not mean squaring each term and leaving out the product between them.

For example,

(x+4)^2

means

(x+4)(x+4)

. Multiplying each term in the first pair by each term in the second pair gives

x^2+4x+4x+16

. Combining the like terms gives

x^2+8x+16

.

(x+p)^2=x^2+2px+p^2

Multiply every term when using the distributive property.
Combine only terms with the same variable part.
A squared binomial includes a middle term.

2. What the two forms show

Vertex form is written as

y=a(x-h)^2+k

. The letters

a

,

h

, and

k

stand for numbers. The vertex is the turning point of the quadratic graph, and its coordinates in this form are

(h,k)

. For this conversion, focus on the expression and the operations shown.

Standard form is written as

y=ax^2+bx+c

. Here,

a

is the coefficient of

x^2

,

b

is the coefficient of

x

, and

c

is the constant term. A coefficient is a number multiplying a variable. A constant has no variable.

To convert, expand the square first. Then multiply by the outside factor

a

. Finally, add or subtract

k

and collect like terms. This order helps keep the signs and terms organized.

The sign inside the brackets matters. In

(x-h)^2

, the squared binomial expands to

x^2-2hx+h^2

. The term

h^2

is positive because it comes from squaring

h

. In

(x+h)^2

, the middle term is positive instead.

(x-h)^2=x^2-2hx+h^2

Vertex form: $y=a(x-h)^2+k$ .
Standard form: $y=ax^2+bx+c$ .
Keep track of the sign inside the brackets when expanding.

3. A visual guide to the conversion

The table shows what each part of vertex form contributes after expansion. It is a guide to the steps. The outside factor multiplies all three terms produced by expanding the square.

After multiplication, the squared term and linear term are in the order used by standard form. If there is an additional number outside the brackets, combine it with the constant.

a(x-h)^2+k=ax^2-2ahx+ah^2+k

Expand the brackets before multiplying by the outside factor.
The outside factor applies to every expanded term.
The final constant includes the added or subtracted number.

4. Apply the method

Converting forms is useful when you need the same quadratic written in a way that highlights different features. Vertex form displays the vertex directly. Standard form displays the coefficients of

x^2

,

x

, and the constant. The conversion does not change the quadratic; it rewrites it using expansion and simplification.

When checking your work, substitute a convenient value for

x

into both expressions. If they are equivalent, they give the same result for that value. A check is helpful, but it does not replace careful expansion.

Use expansion to change the form without changing the expression's value.
A sign or multiplication error can change the quadratic.
Check the middle term and the final constant.

What each part contributes

Part of vertex form	After expanding	After multiplying by $a$
$(x-h)^2$	$x^2-2hx+h^2$	$ax^2-2ahx+ah^2$
$+k$	Added after expansion	Combine with the constant term

Worked example

Convert a quadratic with a positive outside factor

Convert

y=2(x-3)^2+5

to standard form.

Expand the square
Write the squared binomial as its expanded equivalent. The middle term is negative because the bracket contains $x-3$ .
$(x-3)^2=x^2-6x+9$
Multiply every term
The factor $2$ is outside the brackets, so it multiplies the squared expression's three terms.
$2(x^2-6x+9)=2x^2-12x+18$
Add the remaining constant
The original expression also has $+5$ . Combine it with $18$ , since both are constants.
$2x^2-12x+18+5=2x^2-12x+23$

Answer: The standard form is

y=2x^2-12x+23

.

Check: For

x=3

, the original gives

y=5

. The converted form gives

18-36+23=5

, so both forms agree for this value.

Common mistakes and how to avoid them

Writing

(x-3)^2

as

x^2+9

and leaving out the middle term.

Correction: Multiply

(x-3)

by

(x-3)

. The two products involving one

x

and one

-3

combine to make

-6x

.

Multiplying the outside factor by only the first term.

Correction: Use the distributive property on every term inside the expanded expression. For example,

2

must multiply the squared term, the linear term, and the constant.

Treating the sign inside the brackets as the sign of the final constant.

Correction: Square the constant from the brackets. For example,

(-3)^2=9

. The sign of the middle term still depends on the sign inside the brackets.

Adding the outside constant before expanding and losing track of terms.

Correction: Expand and multiply first. Then combine the outside constant with the resulting constant term.

Lesson summary

Start with vertex form, $y=a(x-h)^2+k$ .
Expand the squared binomial, including its middle term.
Multiply every expanded term by $a$ .
Combine the resulting constant with $k$ to write standard form, $y=ax^2+bx+c$ .
Check the signs and confirm that the original and converted expressions give the same value for a chosen input.

Check your understanding

Question 1

Which expression is the standard form of

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

?

$y=3x^2+12x+11$
$y=3x^2+6x+11$
$y=3x^2+12x+13$
$y=3x^2-12x+11$

Show answer and explanation

y=3x^2+12x+11

Expand

(x+2)^2

to get

x^2+4x+4

. Multiply each term by

3

to get

3x^2+12x+12

, then subtract

1

to get

3x^2+12x+11

.

Question 2

When converting

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

, what is the coefficient of

x

in standard form?

$-4$
$4$
$-2$
$2$

Show answer and explanation

4

The squared binomial expands to

x^2-2x+1

. Multiplying by

-2

gives

-2x^2+4x-2

. Adding

4

gives

-2x^2+4x+2

, so the coefficient of

x

is

4

.

Key terms

Quadratic: An expression or function whose highest power of the variable is two.
Vertex form: A way to write a quadratic as $y=a(x-h)^2+k$ .
Standard form: A way to write a quadratic as $y=ax^2+bx+c$ .
Coefficient: A number multiplying a variable or variable term.
Like terms: Terms with the same variable part, such as $5x$ and $-2x$ .

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation A2.7. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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