Investigate transformations of quadratic vertex form

Ontario Grade 11 Mathematics

Mathematical Models

See how changes in a rule move and reshape its parabola

A quadratic graph is a parabola. Its position and shape depend on the numbers in its rule. This lesson investigates those changes using vertex form, tables, and graphs. A useful prerequisite is squaring a number: multiply the number by itself. For example,

(-2)^2=4

. You will use this skill to find outputs from quadratic rules.

What you will learn

Recognize quadratic vertex form and identify its vertex.
Describe how changing each parameter moves or reshapes a parabola.
Use a table or graph to investigate transformations.
Write a vertex-form rule from a vertex and a stated opening direction and width.

1. Start with the parent parabola and vertex form

The parent parabola is the basic graph with rule

y=x^2

. A graph shows input-output pairs that follow a rule. For example, when

x=2

, the output is

4

.

A quadratic in vertex form is written as

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

. The vertex is the turning point of the parabola. In this form, its coordinates are

(h,k)

. The axis of symmetry is the vertical line through the vertex. It divides the parabola into two matching sides.

To read the horizontal coordinate, compare the bracket with

x-h

. For example,

x-3

gives

h=3

. The bracket

x+3

is the same as

x-(-3)

, so it gives

h=-3

. The sign inside the bracket may look opposite to the value of

h

:

h=3

places the vertex three units right of the parent vertex, while

h=-3

places it three units left.

The number

k

gives the vertical coordinate of the vertex. A positive

k

moves the vertex up from the parent vertex, and a negative

k

moves it down. Changing

h

or

k

moves the vertex without changing the basic width or opening direction when

a

stays the same.

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

The parent rule is $y=x^2$ , with vertex $(0,0)$ .
For $y=a(x-h)^2+k$ , the vertex is $(h,k)$ and the axis is $x=h$ .
Changing $h$ moves the vertex horizontally; changing $k$ moves it vertically.

2. Investigate how the value of a changes the graph

The value of

a

affects the opening direction and width. If

a

is positive, the parabola opens upward. If

a

is negative, it opens downward.

The size of

a

, without its sign, affects width. When the size of

a

is greater than

1

, the graph is narrower than the parent. When it is between

0

and

1

, the graph is wider. For example, at an input one unit from the vertex,

y=2x^2

has an output twice as far from zero as

y=x^2

. This vertical stretch makes the graph narrower. The rule

y=0.5x^2

has outputs half as far from zero, so the graph is wider.

A negative value of

a

also describes a reflection. A reflection is a flip of a graph across a line. To understand the line here, compare

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

with

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

when the first value of

a

is positive. The vertex stays at

(h,k)

, but points above the vertex move the same distance below it. The flip is across the horizontal line through the vertex,

y=k

—not necessarily across the x-axis. When

k=0

, that line is the x-axis.

Investigate one change at a time. Keep two of

a

,

h

, and

k

fixed while changing the third. A table or graph can show whether the observed movement and shape agree with the rule.

|a|>1\text{ narrower};\quad 0<|a|<1\text{ wider}

Positive $a$ opens upward; negative $a$ opens downward.
If $|a|>1$ , the graph is narrower; if $0<|a|<1$ , it is wider.
A negative $a$ flips the graph across the horizontal line through its vertex.

3. Use tables and graphs to investigate symmetry

A table can help check a transformation. Choose inputs on both sides of the vertex. Inputs the same distance from the axis of symmetry have equal outputs. This happens because the squared part of the rule is the same for both inputs.

For

y=(x-1)^2+2

, the vertex is

(1,2)

. The inputs

0

and

2

are each one unit from

1

, and both give an output of

3

. The inputs

-1

and

3

are each two units from

1

, and both give an output of

6

. These pairs show symmetry around the axis

x=1

.

A graphing tool can display several rules on the same coordinate grid. Change one parameter at a time. Observe the vertex, opening direction, and width. Make sure the viewing window includes the vertex; otherwise, an important feature may be hidden.

When describing a transformation, name the vertex and describe the shape. For example, say that the vertex moved right and down, that the graph opens downward, and that it is wider than the parent. A table or graph can help check each part of that description.

x=h-d and x=h+d \Rightarrow y_1=y_2

Inputs equally far from the axis of symmetry have equal outputs.
Use a table or graph to check what a rule predicts.
Describe position, opening direction, and width.

4. Build a rule from graph features

If you know the vertex and the vertical stretch or reflection, you can write a rule in vertex form. Place the vertex coordinates in

h

and

k

. Then choose

a

to match the opening direction and width.

For a vertex at

(-2,3)

, the bracket is

x-(-2)

, or

x+2

, and the final term is

+3

. A downward-opening graph needs a negative value of

a

. These choices describe the graph’s position and shape.

Check the rule by finding its vertex and testing nearby inputs. At the vertex, the squared part is zero. Inputs equally far to either side should give equal outputs. These checks connect the symbols to the graph and can reveal a sign error.

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

Use the vertex coordinates for $h$ and $k$ , taking care with the sign in the bracket.
Use the sign and size of $a$ to match opening direction and width.
Check the vertex and symmetry with nearby inputs.

Compare outputs near the vertex

Input $x$	$y=x^2$	$y=2x^2$	$y=0.5x^2$
$-2$	$4$	$8$	$2$
$-1$	$1$	$2$	$0.5$
$0$	$0$	$0$	$0$
$1$	$1$	$2$	$0.5$
$2$	$4$	$8$	$2$

Worked example

Compare a transformed rule with the parent

Describe the vertex, axis of symmetry, opening direction, and width of

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

. Find the outputs for inputs one unit to either side of the vertex.

Read the vertex
Compare the rule with vertex form. The bracket is $x-3$ , so $h=3$ , and the final constant gives $k=1$ . The vertex is $(3,1)$ . The axis is the vertical line through it.
$(h,k)=(3,1),\quad x=3$
Describe the shape
Here $a=-2$ . Its negative sign means the parabola opens downward. Its size is greater than $1$ , so the graph is narrower than the parent parabola.
$a=-2$
Check points around the vertex
The inputs one unit to either side of $3$ are $2$ and $4$ . In each case, the squared bracket is $1$ , so the output is $1-2=-1$ . The matching outputs confirm symmetry.
$y(2)=y(4)=-1$

Answer: The vertex is

(3,1)

and the axis of symmetry is

x=3

. The parabola opens downward and is narrower than

y=x^2

. The outputs at inputs

2

and

4

are both

-1

.

Check: Because the graph opens downward, its vertex is its highest point. The inputs

2

and

4

are equally far from the axis

x=3

, and their outputs match.

Worked example

Write a rule from graph features

Write a vertex-form rule for a parabola with vertex

(-2,3)

that opens downward and has a vertical stretch factor of

2

. Check two inputs equally far from the vertex.

Use the vertex coordinates
The vertex gives $h=-2$ and $k=3$ . In the bracket, $x-h$ becomes $x-(-2)$ , which is the same as $x+2$ .
$(h,k)=(-2,3)$
Choose a to match the shape
A vertical stretch factor of $2$ means the size of $a$ is $2$ . Downward opening needs a negative sign, so choose $a=-2$ . Substituting these values gives the rule.
$y=-2(x+2)^2+3$
Check symmetric inputs
The inputs $-3$ and $-1$ are each one unit from the vertex’s horizontal coordinate, $-2$ . For both, the squared bracket is $1$ , so both outputs are $3-2=1$ .
$y(-3)=y(-1)=1$

Answer: One suitable rule is

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

. It has vertex

(-2,3)

, opens downward, and has a vertical stretch factor of

2

.

Check: The squared part is zero at the vertex, giving the stated coordinates. Equal outputs at

-3

and

-1

confirm symmetry around

x=-2

.

Common mistakes and how to avoid them

Reading

x+4

as a shift right four units.

Correction: Rewrite it as

x-(-4)

. The vertex has horizontal coordinate

-4

, so it is shifted left four units.

Using

a

only to describe whether the parabola opens up or down.

Correction: The sign of

a

gives the opening direction, and its size also affects the width.

Forgetting the final

+k

term when writing the rule.

Correction: The value of

k

gives the vertex’s vertical coordinate and shifts the graph vertically.

Expecting different outputs for inputs equally far from the axis of symmetry.

Correction: For a rule in vertex form, those inputs have the same squared part and therefore the same output.

Lesson summary

Vertex form is $y=a(x-h)^2+k$ . Its vertex is $(h,k)$ and its axis of symmetry is $x=h$ .
Changing $h$ shifts the vertex horizontally, and changing $k$ shifts it vertically.
The sign of $a$ sets the opening direction. Its size affects width, and a negative value flips the graph across the horizontal line through the vertex.
Tables and graphs help check position, shape, and symmetry.

Check your understanding

Question 1

What is the vertex of

y=3(x+1)^2-4

?

$(1,-4)$
$(-1,-4)$
$(-1,4)$
$(3,-4)$

Show answer and explanation

(-1,-4)

The bracket

x+1

is

x-(-1)

, so

h=-1

. The vertical coordinate is

k=-4

.

Question 2

Which description matches

y=-0.5(x-2)^2+5

?

It opens upward and is narrower than the parent.
It opens downward and is wider than the parent.
It opens downward and is narrower than the parent.
It opens upward and is wider than the parent.

Show answer and explanation

It opens downward and is wider than the parent.

The negative value of

a

means downward opening. Its size is less than

1

, so the graph is wider than the parent.

Question 3

For

y=(x-4)^2-2

, which pair of inputs must give equal outputs because they are equally far from the axis?

$3$ and $5$
$2$ and $5$
$4$ and $5$
$2$ and $6$

Show answer and explanation

3

and

5

The axis is

x=4

. Inputs

3

and

5

are each one unit from it, so their outputs match.

Key terms

Quadratic: A rule whose graph is a parabola, such as $y=x^2$ .
Vertex: The turning point of a parabola.
Axis of symmetry: The vertical line through the vertex that divides a parabola into matching sides.
Transformation: A change in a graph’s position or shape.
Vertical stretch: A change that moves outputs farther from the horizontal line through the vertex, making the parabola narrower.
Reflection: A flip across a line. Changing a positive value of $a$ to its negative, while keeping $h$ and $k$ fixed, flips the parabola across the horizontal line $y=k$ through its vertex.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation A1.3. 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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