Investigate transformations in vertex form

Ontario Grade 11 Mathematics

Quadratic Functions

How changes in an equation move or reshape a graph

A graph can change in a predictable way when numbers in its equation change. For example, moving a quadratic graph to the right also moves its vertex to the right. In this lesson, you will connect these changes to equations and graphs. The main focus is how vertex form makes transformations visible. We will also connect the same idea to exponential and sine functions without building a more complicated sine model.

What you will learn

Describe how changes to an equation affect a graph’s position, direction, or shape.
Read the vertex and transformations from a quadratic in vertex form.
Connect transformations of quadratic, exponential, and sine graphs to equations and key features.
Use a graph or graphing technology to check a transformation.

1. Prerequisite bridge: points and graph changes

A function pairs each input with one output. Its graph is a picture of those input-output pairs. The parent function is the basic graph before transformations are applied. For a quadratic, a common parent function is

y=x^2

. Its vertex is the turning point of the graph.

A transformation changes a graph’s position or shape. A translation slides a graph without changing its shape. A reflection flips it. A vertical stretch makes it steeper, while a vertical compression makes it less steep.

A useful point rule connects an old graph to a new one. If every old point

(x,y)

moves to

(x+h,y+k)

, the graph shifts right by

h

and up by

k

. A negative shift moves left or down. Keep track of whether a change is inside or outside the function: an outside change affects output values, while an inside change affects input values.

(x,y)\rightarrow(x+h,y+k)

The vertex is the turning point of a quadratic graph.
Outside changes affect vertical position; changes inside the function affect horizontal position.
A graph can be checked by comparing its important points before and after a change.

2. Quadratic transformations in vertex form

Vertex form is a way to write a quadratic so its vertex and several transformations can be read directly. In

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

, the vertex is

(h,k)

. The number

a

controls the direction and vertical steepness.

If

a

is positive, the parabola opens upward. If

a

is negative, it opens downward. If the absolute value of

a

is greater than

1

, the graph is vertically stretched compared with

y=x^2

. If it is between

0

and

1

, the graph is vertically compressed.

The signs inside the brackets can seem reversed. The expression

x-h

shifts the graph right by

h

, while

x+h

shifts it left by

h

. The value outside the square,

k

, shifts the graph up when positive and down when negative.

You can investigate these effects by changing one value at a time in a graphing tool. Keep the other values fixed. Observe the vertex, whether the graph opens up or down, and how wide it looks. This makes it easier to tell which part of the equation caused each change.

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

In $y=a(x-h)^2+k$ , the vertex is $(h,k)$ .
The sign of $a$ determines whether the parabola opens up or down.
The size of $a$ affects the vertical stretch or compression.

3. The same transformation idea in exponential and sine graphs

Transformations are not limited to parabolas. An exponential parent function such as

y=2^x

has a curve that increases as

x

increases. A vertical translation moves the whole curve up or down. A horizontal translation changes where the curve’s key positions occur. For instance, replacing

x

with

x-h

shifts the graph right by

h

.

A sine graph repeats a wave pattern. The parent function

y=\sin x

is a repeating curve. A vertical translation moves the wave up or down, and a reflection in the horizontal axis flips it. A horizontal translation moves the wave left or right. These changes can be investigated by comparing the parent graph with a graph where one change has been made.

For all three function types, use the same investigation routine: identify the parent graph, change one part of the equation, and compare matching features. For a quadratic, track the vertex. For an exponential graph, compare its position and direction of change. For a sine graph, compare the wave’s position and orientation. A graphing tool can help, but the equation still explains the change.

y=2^{x-h}+k

Translations can also be investigated on exponential and sine graphs.
Change one part of the equation at a time to identify its effect.
Use features suited to each graph: a vertex, an exponential curve’s position, or a sine wave’s position and orientation.

4. Guided example and application

A quadratic model can describe the height of an object over time when its path forms a parabola. In vertex form, the vertex identifies the maximum or minimum point. The other values describe how the graph is positioned and whether it opens up or down.

In the example, compare the given quadratic with the parent graph

y=x^2

. Read the vertex first, then interpret the sign and size of the coefficient. Finally, check a nearby point to confirm the direction of the shift and the effect on the graph.

Read the vertex before describing the remaining transformations.
A point from the equation can confirm your interpretation of the graph.

Reading quadratic vertex form

Part of equation	Graph effect
$h$ in $(x-h)$	Shifts the graph right by $h$ ; a negative $h$ shifts it left.
$k$ outside the square	Shifts the graph up if positive and down if negative.
$a>0$ or $a<0$	Positive opens upward; negative opens downward.
$\|a\|>1$ or $0<\|a\|<1$	Greater than $1$ stretches; between $0$ and $1$ compresses.

Worked example

Read and check a quadratic transformation

Describe the transformations and vertex of

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

compared with

y=x^2

. Find the output when

x=4

.

Find the vertex
Match the equation to vertex form. The expression inside the brackets is $x-3$ , so the graph shifts right by $3$ . The outside value is $1$ , so it shifts up by $1$ . The vertex is $(3,1)$ .
$(h,k)=(3,1)$
Describe the shape
The coefficient is negative, so the parabola opens downward. Its absolute value is $2$ , which means it is vertically stretched by a factor of $2$ compared with the parent graph.
$a=-2$
Check a nearby point
Substitute $x=4$ . Since this input is one unit to the right of the vertex, the squared part is $1$ . The output is $-1$ , which is one unit below the vertex, consistent with the graph opening downward.
$y=-2(4-3)^2+1=-1$

Answer: The graph shifts right

3

, shifts up

1

, reflects across the horizontal axis, and is vertically stretched by a factor of

2

. Its vertex is

(3,1)

, and when

x=4

,

y=-1

.

Check: At the vertex input

x=3

, the squared part is zero, so the output is

1

. This confirms the vertex

(3,1)

.

Common mistakes and how to avoid them

Reading

x-3

as a shift left by

3

.

Correction: The expression

x-3

shifts the graph right by

3

. Check the vertex to confirm the direction.

Thinking a negative coefficient moves the graph down.

Correction: A negative coefficient in front of the squared part reflects the parabola and makes it open downward. The outside value

k

controls vertical translation.

Calling a graph wider when

|a|>1

.

Correction: A value with absolute value greater than

1

makes the parabola narrower through a vertical stretch. A value between

0

and

1

makes it wider through a vertical compression.

Changing several equation values at once and guessing which caused the graph change.

Correction: Change one value at a time, then compare the graph with the parent function.

Lesson summary

Transformations change a graph’s position, direction, or shape.
Quadratic vertex form makes the vertex and main transformations easy to identify.
For exponential and sine graphs, compare the transformed graph with its parent and track suitable features.
A graphing tool can support an investigation, while the equation explains the result.

Check your understanding

Question 1

For

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

, what is the vertex?

$(-2,-4)$
$(2,-4)$
$(-2,4)$
$(3,-4)$

Show answer and explanation

(-2,-4)

The expression

x+2

is

x-(-2)

, so the horizontal coordinate is

-2

. The outside value is

-4

, giving the vertex

(-2,-4)

.

Question 2

Compared with

y=x^2

, what does a negative value of

a

do in

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

?

It makes the parabola open downward.
It shifts the vertex left.
It shifts the graph up.
It makes the parabola open upward.

Show answer and explanation

It makes the parabola open downward.

A negative coefficient reflects the parabola so that it opens downward. The values

h

and

k

determine the vertex position.

Key terms

Parent function: The basic function used as a starting graph before transformations.
Transformation: A change to a graph’s position, direction, or shape.
Translation: A slide of a graph that does not change its shape.
Vertex: The turning point of a quadratic graph.
Vertical stretch or compression: A change that makes a graph steeper or less steep by multiplying its output values.
Reflection: A flip of a graph across a line, such as the horizontal axis.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation A2.5. 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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