Learning activity 2.3: Exploring transformations of quadratic functions

Inuit culture and parabolas

An individual in the midst of making an igloo using modern shovels and ice picks.

Igloos are traditional Inuit homes made up of snow blocks. They are in the shape of an archeddome and are designed with the knowledge that hot air rises and cold air sinks. Inside the dome hot air is produced from human bodies and the qulliq (seal oil lamp) eventually becoming trapped in the Igloo. Cold air sinks and collects at the entrance since it is the lowest point of the dwelling. The ventilation holes around the dome release the carbon dioxide produced inside.

Meaning of transformations

Transformations are operations performed on functions to change the position or shape of the associate curves or lines. In other words, the meaning of transformation is "to change from one place, state, form, or appearance to another". When we take a function and tweak its value so that its graph is moved to another spot on the axis system, yet remains recognizably the same graph, we are said to be "transforming" the function.

Explore this!

Let us examine the detailed steps involved in this method in the following video.

The following are some of the most important ideas from the video.

Key deas

Quadratic functions can be written in the form $g (x) = a {(x - h)}^{2} + k$ .
Each of the constants a, h, and k changes the position and/or shape of the graph of $f (x) = {(x)}^{2}$ .

Need to know

Changing the values of h and k changes the position of the parabola and, as a result, the locations of the vertex and the axis of symmetry. The new parabola is congruent to the parabola $f (x) = {(x)}^{2}$ .
Changing the value of a can change the shape of the parabola, as well as the direction in which the parabola opens. The new parabola is not congruent to the parabola $f (x) = {(x)}^{2}$ when a ≠ 1 or −1.

Graphing tool

Throughout the remainder of this course, you’ll use the following graphing app, GeoGebra, provided for several investigations involving quadratic functions. Explore the following instructions to learn how to use the app properly. As practice, graph the function: $y = 3 (x - 4)^{2} + 7$

Graphing quadratic functions

You can use this applet to graph and compare quadratic functions for Investigations and Support Questions.

To enter equations

for exponents, use the ^ caret;
for multiplication, use the * asterisk;
for division, use the / slash;
ensure you use brackets correctly;
press the [Enter] key.

For example, to graph y = 3(x – 4)² + 7, enter y=3*(x-4)^2+7.

GeoGebra tricks

You can enter two functions, then add or subtract or multiply or divide them. Try the following equations:

f(x)=x^2+10
g(x)=x+5
h(x)=f(x)-g(x)

Created with GeoGebra (Opens in new window)

Press the start button if you wish to use the GeoGebra graphing tool. The GeoGebra tool will open in a new window, and can be used throughout this part of the learning activity.

Blank Cartesian plane with x and y axes moving from intervals of 2.

Start

We will use function notation when naming the graphs to make it easier to identify specific graphs in the investigations.

You can use any of the quadratic functions from earlier in the course (or make up your own) for further practice using the graphing app.

Investigation 1: $a > 1$

In this activity we will examine transformations. A transformation is a modification on the output values of a function because of a change to the parameters in the equation. In the equation $y = a (x - h)^{2} + k$ , x and y are variables and a,h and k are parameters.

A building with dynamic parabolic arches.

Complete the following tables of values for the two functions $y = x^{2}$ and $y = 2 x^{2}$ .

When you’re finished, compare your answers to the suggested ones.

$y = x^{2}$

$x$	$y$
-2	4
-1	1
0	0
1	1
2	4

$y = 2 x^{2}$

$x$	$y$
-2	8
-1	2
0	0
1	2
2	8

Notebook

When you have completed using the graphing app, use your notebook to complete the following questions. Check your answers with the suggested answers provided.

How do you think the $y$ -values between the two functions are related?

The $y$ -values for the second equation are double those of the first equation.

Predict what the $y$ -values of $y = 3 x^{2}$ might be in comparison to those of $y = x^{2}$ .

The $y$ -values would be tripled.

To check if your prediction was correct, create a table of values for $y = 3 x^{2}$ using the same $x$ -values as earlier.

Now use the following graphing app to graph $y = x^{2}$ . Then press enter and graph $y = 2 x^{2}$ together on the same screen.

Press the start button if you wish to use the GeoGebra graphing tool. The GeoGebra tool will open in a new window, and can be used throughout this part of the learning activity.

Start

Notebook

When you have completed using the graphing app, use your notebook to complete the following questions. Check your answers with the suggested answers provided.

Describe the similarities between the graphs of $y = x^{2}$ and $y = 2 x^{2}$

They both open the same direction. They both have the same vertex and it is on the origin $(0, 0)$ .

Describe the differences.

The graph of $y = 2 x^{2}$ is skinnier than the other graph.

How does changing the $a$ -value to $2$ affect the graph of of $y = {a x}^{2}$ ?

It stretches it vertically and becomes skinnier.

Predict what the graph of $y = {3 x}^{2}$ might be like in comparison to these graphs.

It will stretch even more.

To check if your prediction was correct, add the graph of $y = {3 x}^{2}$ to your graph using the graphing app.

Provide a conclusion about how the a-value, for $a > 1$ in $y = a x^{2}$ transforms the graph of $y = x^{2}$ .

When $a > 1$ , it vertically stretches the graph by that a value.

Investigation 2: $0 < a < 1$

Complete the tables of values for following two functions. When you’re finished, compare your answers with those provided.

$y = x^{2}$ $y = \frac{1}{2}$

$x$	$y$
$- 4$	$16$
$- 2$	$4$
$0$	$0$
$2$	$4$
4	$16$

$y = \frac{1}{2} x^{2}$

$x$	$y$
$- 4$	$8$
$- 2$	$2$
$0$	$0$
$2$	$2$
$4$	$8$

How are the $y$ -values related?

The $y$ values of the second function are half of the $y$ values of the first function.

Use the following graphing tool to graph $y = x^{2}$ and $y = \frac{1}{2} x^{2}$ together.

Press the start button if you wish to use the GeoGebra graphing tool. The GeoGebra tool will open in a new window, and can be used throughout this part of the learning activity.

Start

Notebook

When you have completed using the graphing app, use your notebook to complete the following questions. Check your answers with the suggested answers provided.

Describe the similarities.

They both open the same direction. They both have the same vertex and it is on the origin $(0, 0)$ .

Describe the differences.

The first graph is skinner than the second graph.

How does changing the $a$ -value to $\frac{1}{2}$ affect the graph of $y = x^{2}$ .

The $\frac{1}{2}$ value of $a$ , compresses the parabola vertically.

Predict what the graph of $y = \frac{1}{4} x^{2}$ might be like in comparison to these graphs.

It would be compressed even more.

To check if your prediction was correct, add the graph of $y = \frac{1}{4} x^{2}$ to your graph using the graphing app.

Provide a conclusion about how the a value, for $0 < a < 1$ in $y = a x^{2}$ transforms the graph of $y = x^{2}$ .

When $0 < a < 1$ , it vertically compresses the graph by $a$ .

Investigation 3: $a < - 1$

Use the following graphing application to graph $y = x^{2}$ and $y = {- x}^{2}$ on the same grid.

Press the start button if you wish to use the GeoGebra graphing tool. The GeoGebra tool will open in a new window, and can be used throughout this part of the learning activity.

Depiction of a graphing tool used for investigating transformations.

Start

Notebook

When you have completed using the graphing app, use your notebook to complete the following questions. Check your answers with the suggested answers provided.

Describe the similarities between the graphs of $y = x^{2}$ and $y = {- x}^{2}$ .

Same vertex $(0, 0)$ , both graphs are the same width.

Describe the differences.

The graph of $y = {- x}^{2}$ opens down.

How does changing the $a$ -value to $- 1$ affect the graph of $y = x^{2}$ ?

It flips (reflects) the graph over the $x$ -axis.

Predict what the graph of $y = {- 2 x}^{2}$ and $y = {- 3 x}^{2}$ might be like in comparison to these graphs.

The negative ‘ $a$ ’ value tells us it’s reflected about the $x$ -axis. Since $a < - 1$ , the graph would be stretched vertically by $2$ for the first graph and $3$ for the second graph.

To check if your prediction was correct, add the graphs of $y = {- 2 x}^{2}$ and $y = {- 3 x}^{2}$ to your graph using the graphing app.

Provide a conclusion about how the value $a$ , where $a < - 1$ in $y = {a x}^{2}$ transforms the graph of $y = x^{2}$ .

When a is less than $- 1$ , the graph is reflected (flipped) about the $x$ -axis and then vertically stretched by the positive value of the number.

For example: $a = - 2$ , would be reflected about the $x$ -axis and vertically stretched by $2$ .

Investigation 4: $- 1 < a < 0$

Add $y = - \frac{1}{2} x^{2}$ and $y = - \frac{1}{4} x^{2}$ to your graph from Investigation $1$ . (Investigation 1 graphs $y = x^{2}, y = 2 x^{2}$ )

Press the start button if you wish to use the GeoGebra graphing tool. The GeoGebra tool will open in a new window, and can be used throughout this part of the learning activity.

Start

Notebook

When you have completed using the graphing app, use your notebook to complete the following question. Check your answers with the suggested answers provided.

Provide a conclusion about how the value $a$ , where $- 1 < a < 0$ in $y = {a x}^{2}$ transforms the graph of $y = x^{2}$ .

When a is greater than $- 1$ and less than $0$ , the graph is reflected (flipped) about the $x$ -axis and then vertically compressed by the positive value of the number.

For example: $a = - \frac{1}{2}$ , would be reflected about the $x$ -axis and vertically compressed by $\frac{1}{2}$ .

To solidify what you've learned in the investigations, explore the following graph application more with sliders show how different $a$ values affect the graph of $y = a x^{2}$

Write a short paragraph explaining what you explored and how the a values affect the graph. What happens to the graph when the value increases? Decreases? Submit your answer to your portfolio for feedback.

Start

Sketching parabolas of $y = a^{2}$

An axis of symmetry is an imaginary line that cuts a parabola in half and passes through the vertex. Let’s compare the properties of the graph of $y = x^{2}$ to the graph of $y = {- x}^{2}$ . Sketch the graphs with technology or in your notebook. Compare your answers with those provided.

$y = x^{2}$

Graph properties for $y = x^{2}$

Concavity

The graph opens upward, so it is concave up.

Vertex

The lowest point $(0, 0)$ is the vertex.

Maximum or minimum value

The $y$ -value at the vertex, $y = 0$ , is the minimum value of the graph.

Domain

$D = {x \in R}$ since all input values can be real numbers.

Range

$R = {y \in R | y \geq 0}$ since the lowest $y$ -value is $0$ (at the vertex) and all other $y$ -values are above $0$ (because the graph is concave up).

Axis of symmetry

Each side of the parabola is reflected (or symmetrical) in the vertical line $x = 0$ , called the axis of symmetry. The axis of symmetry cuts the parabola in half and passes through the vertex.

$y = {- x}^{2}$

Graph properties for $y = {- x}^{2}$

Concavity

The graph opens downward, so it is concave down.

Vertex

The highest point $(0, 0)$ is the vertex.

Maximum or minimum value

The $y$ -value of the vertex, $y = 0$ , is the maximum value of the graph.

Domain

$D = {x \in R}$ since all input values can be real numbers.

Range

$R = {y \in R | y \leq 0}$ since the highest $y$ -value is $0$ (at the vertex) and all other $y$ -values are below $0$ (because the graph is concave down).

Axis of symmetry

Each side of the parabola is reflected (or symmetrical) in the vertical line $x = 0$ , called the axis of symmetry. The axis of symmetry cuts the parabola in half and passes through the vertex.

Did you notice that the graph of $y = {- x}^{2}$ is obtained by reflecting the graph of $y = x^{2}$ in the $x$ -axis?

Notice that both functions have the same vertex, domain, and axis of symmetry.

Symmetrical points

The axis of symmetry cuts a parabola in half. Any two points that are directly across from each other on the parabola can be joined by a horizontal line that is perpendicular to the axis of symmetry. These points will have the same $y$ -values and are equidistant from the axis of symmetry.

Let’s examine the following graph of $y = 0.5 x^{2}$ .

For the graph, the points A and B are symmetrical points. They have the same $y$ -coordinate. These two points are the same distance (equidistant) from the axis of symmetry.

The same is true for the points C and D, and any other points that are on opposite sides of the axis of symmetry.

Practice on role of $a$ in $y = a x^{2}$

Describe the transformations that are applied to the graph of $y = x^{2}$ to obtain the graph of $y = - \frac{3}{4} x^{2}$ .

The value of the coefficient $a = - \frac{3}{4}$ is between $- 1$ and $0$ . The parabola $y = x^{2}$ is vertically compressed by $\frac{3}{4}$ and reflected in the $x$ -axis.

Describe the following characteristics for the graph $y = - \frac{3}{4} x^{2}$ . When you’re finished, compare your answers with the suggested ones.

Vertex

The vertex is $(0, 0)$ because this parabola is a vertical compression of $y = x^{2}$ . Compressing the graph does not affect the vertex.

Axis of symmetry

It has the same axis of symmetry as $y = x^{2}$ , that is, $x = 0$ (or the $y$ -axis).

Concavity

The parabola is concave down because it is a reflection in the $x$ -axis.

Domain

$D = {x \in R}$

Range

$R = {y \in R | y \leq 0}$

Minimum or maximum value

The parabola is concave down. The vertex $(0, 0)$ is the maximum point and so the maximum value is $y = 0$ .

Two symmetrical points

Any two symmetrical points will be on opposite sides of the axis of symmetry, which is the $y$ -axis. You may choose the positive and negative of any $x$ -value, such as $\pm 2$ . Find the corresponding $y$ -value by substituting into the equation.

Let $x = 2$ , then $y = - \frac{3}{4} (2) 2 = - 3$ .

Let, $x = - 2$ then $y = - \frac{3}{4} (- 2) 2 = - 3$ .

Notice that the $y$ -values are indeed equal.

So $(2, - 3)$ and $(- 2, - 3)$ are two symmetrical points.

Notebook

Using your notebook and the information you discovered above— sketch a graph of the function. When you’re finished, compare your sketch with the suggested answer.

Represent the equations of two different quadratic functions that represent the transformations for each of the following cases:

$y = x^{2}$ is stretched vertically.

All the equations will have the form $y = {a x}^{2}$

There are many possible answers.

A vertical stretch occurs when $a > 1$

Two possible equations are $y = {2 x}^{2}$ and $y = {7 x}^{2}$

$y = x^{2}$ is compressed vertically and reflected in the $x$ -axis.

All the equations will have the form $y = a x^{2}$

There are many possible answers.

A vertical compression and reflection in the $x$ -axis occurs when $- 1 < a < 0$

Two possible equations are $y = {- \frac{2}{9} x}^{2}$ and $y = {- \frac{1}{8} x}^{2}$

This next exercise shows how you can use points on the graph of a parabola to find the value of $a$ in the transformed equation.

Given this graph of a parabola (the blue curve), answer the following questions:

Describe the transformation(s) that have been applied to the graph of $y = x^{2}$ (the green curve).

The graph of $y = x^{2}$ has been vertically compressed and reflected in the $x$ -axis, so the a-value in the transformed equation $y = {a x}^{2}$ satisfies $- 1 < x < 0$ .

Determine the equation of the blue parabola.

The two indicated points on the graph are $(4, - 4)$ and $(- 4, - 4)$ . Substitute one of the points into the equation $y = {a x}^{2}$ and solve for $a$ .

Use $(4, - 4)$ . Substitute $x = 4$ and $y = - 4$

$- 4 = a (4)^{2}$

$- 4 = 16 a$

$- \frac{4}{16} = a$

$- \frac{1}{4} = a$

The equation of the blue parabola is $y = - \frac{1}{4} x^{2}$

Investigation 5: $h > 0$

Graph $y = x^{2}$ and $y = {(x - 1)}^{2}$ together. When you’re finished graphing these two functions, continue on to answer some practice questions.

Press the start button if you wish to use the GeoGebra graphing tool. The GeoGebra tool will open in a new window, and can be used throughout this part of the learning activity.

Start

Notebook

When you have completed using the graphing app, use your notebook to complete the following questions. Check your answers with the suggested answers provided.

Compare the two graphs. Describe their similarities.

They both open up. They both are the same width. The vertex has the same $y$ -value.

Describe their differences.

One is moved along the $x$ -axis.

Consider the role of $h$ in $y = a {(x - h)}^{2} + k$ . How does the value $h = 1$ in $y = {(x - 1)}^{2}$ change the graph of $y = x^{2}$ ?

It moves the graph one unit to the right.

Graph the following functions using the following applet:

$y = {(x - 2)}^{2}$

$y = {(x - 3)}^{2}$

$y = {(x - 4)}^{2}$

Submit a copy of your graphs to your portfolio for feedback.

Press the start button if you wish to use the GeoGebra graphing tool. The GeoGebra tool will open in a new window, and can be used throughout this part of the learning activity.

Start

Notebook

When you have completed using the graphing app, use your notebook to complete the following questions. Check your answers with the suggested answers provided.

Compare all the graphs. Describe their similarities.

All are concave up. They all have the same $y$ -value for their vertex. They are all the same width.

Describe their differences.

Each one keeps moving to the right by one unit.

In general, how does the value of $h (h > 0)$ in $y = {(x - h)}^{2}$ transform the graph of $y = x^{2}$ ?

For $h > 0$ in $y = {(x - h)}^{2}$ the graph will be translated $h$ units to the right.

Investigation 6: $h < 0$

You will use the graphing app to answer the following questions. Suppose the value of $h$ is negative, $h = - 1$ . Substituting $h = - 1$ in $y = {(x - h)}^{2}$ results in $y = {(x - (- 1))}^{2}$ , which simplifies to $y = {(x + 1)}^{2}$ .

Think!

Predict what the graph of $y = {(x + 1)}^{2}$ might be like in comparison to the graph $y = x^{2}$ .

It may move to the left since we learned that when $h > 0$ , it moves to the right.

Press the start button if you wish to use the GeoGebra graphing tool. The GeoGebra tool will open in a new window, and can be used throughout this part of the learning activity.

Start

Notebook

When you have completed using the graphing app, use your notebook to complete the following questions. Check your answers with the suggested answers provided.

Was your prediction correct?

Yes the graph was moved one unit to the left.

Add the following functions to your graph:

$y = {(x + 2)}^{2}$
$y = {(x + 3)}^{2}$
$y = {(x + 4)}^{2}$

Compare all the graphs. Describe their similarities.

All are concave up. They all have the same $y$ -value for their vertex. They are all the same width.

Describe their differences.

The graphs continue to be translated one unit to the left.

How does the value $h (h < 0)$ in $y = {(x - h)}^{2}$ transform the graph of $y = x^{2}$ ?

For $h < 0$ in $y = {(x - h)}^{2}$ the graph will be translated $h$ units to the left.

Practice the role of $h$ in $y = {(x - h)}^{2}$

Each of the following graphs represents a horizontal translation of the graph of $y = x^{2}$ . Complete an analysis of the following graph to determine the graph’s equation.

This graph represents a horizontal translation of the graph of $y = x^{2}$ . Complete an analysis of the graph to determine the graph’s equation.

Investigation 7: $k > 0$

Use the graphing app to graph $y = x^{2}$ and $y = x^{2} + 1$ .

Press the start button if you wish to use the GeoGebra graphing tool. The GeoGebra tool will open in a new window, and can be used throughout this part of the learning activity.

Start

Notebook

When you have completed using the graphing app, use your notebook to complete the following questions. Check your answers with the suggested answers provided.

Compare the two graphs. Describe their similarities.

Both are concave up. Both have the same axis of symmetry. Both are the same width.

Describe their differences.

The graphs are shifted up and down.

Consider the role of $k$ in $y = a {(x - h)}^{2} + k$ . How does the value $k = 1$ in $y = x^{2} + 1$ change the graph of $y = x^{2}$ ?

It moves the graph up one unit.

Graph the following functions

$y = x^{2} + 2$
$y = x^{2} + 3$
$y = x^{2} + 4$

Press the start button if you wish to use the GeoGebra graphing tool. The GeoGebra tool will open in a new window, and can be used throughout this part of the learning activity.

Start

Notebook

Using your notebook, compare all the graphs. Describe their similarities. Compare your answers with the suggestions provided.

Compare all the graphs. Describe their similarities.

All are concave up. All have the same width. All have the same axis of symmetry.

Describe their differences.

All are shifted up by their $k$ values.

How does the value of $k (k > 0)$ in $y = x^{2} + k$ transform the graph of $y = x^{2}$ ?

For $k > 0$ in $y = x^{2} + k$ the graph will be translated k units up.

Investigation 8: $k < 0$

Think

Predict what the graph of $y = x^{2} - 1$ might be like in comparison to the graph $y = x^{2}$ .

Translate the graph down $1$ unit.

Now, graph both functions using the graphing app. Was your prediction correct?

Press the start button if you wish to use the GeoGebra graphing tool. The GeoGebra tool will open in a new window, and can be used throughout this part of the learning activity.

Start

Add to the above two functions by entering the following functions.

$y = x^{2} - 2$
$y = x^{2} - 3$
$y = x^{2} - 4$

Notebook

When you have completed using the graphing app, use your notebook to complete the following questions. Check your answers with the suggested answers provided.

Compare all five graphs. Describe their similarities.

All are concave up. All have the same width. All have the same axis of symmetry.

Describe their differences.

All are shifted down by their k values.

How does the value of $k (k < 0)$ in $y = x^{2} + k$ transform the graph of $y = x^{2}$ ?

For $k < 0$ in $y = x^{2} + k$ the graph will be translated $k$ units down.

Putting it all together: $a$ , $h$ , and $k$ in $y = a (x - h)^{2} + k$

Predict what the graph of $y = 2 (x - 4)^{2} + 1$ might be like in comparison to the graph $y = x^{2}$ .

Answers will vary but should address how you think the $a$ , $h$ , and $k$ , values will affect the graph.

Graph both functions using the following graphing app.

Press the start button if you wish to use the GeoGebra graphing tool. The GeoGebra tool will open in a new window, and can be used throughout this part of the learning activity.

Start

Notebook

When you have completed using the graphing app, use your notebook to complete the following questions. Check your answers with the suggested answers provided.

Was your prediction correct?

Stretch vertically by $2$ . Translated to the right $4$ . Translated up $1$ .

Add the function $y = - 2 (x - 4)^{2} + 1$ to your graph. Compare the three graphs. Describe their similarities.

All parabolas, just transformed.

Describe their differences.

Different vertex. Different widths. Direction of opening is different.

How has the graph of $y = x^{2}$ been transformed to get the graph of $y = 2 (x - 4)^{2} + 1$ ?

Vertically stretched by $2$ .

Horizontally translated $4$ units right

Vertically translated $1$ unit up.

How has it been transformed to get the graph of $y = - 2 (x - 4)^{2} + 1$ ?

Reflected about the $x$ axis

Vertically stretched by $2$

Horizontally translated $4$ units right.

Vertically translated $1$ unit up.

Add the following two functions to your graph:

$y = \frac{1}{2} (x - 4)^{2} + 1$
$y = - \frac{1}{2} (x - 4)^{2} + 1$

Press the start button if you wish to use the GeoGebra graphing tool. The GeoGebra tool will open in a new window, and can be used throughout this part of the learning activity.

Start

Notebook

When you have completed using the graphing app, use your notebook to complete the following questions. Check your answers with the suggested answers provided.

Compare your two graphs. Describe their similarities.

Both are compressed. Both have moved right four and up $1$ .

Describe their differences.

They are reflections of each other about the $x$ -axis.

How has the graph of $y = x^{2}$ been transformed to get the graph of $y = \frac{1}{2} (x - 4)^{2} + 1$ ?

Vertically compressed by $\frac{1}{2}$ .

Horizontally translated $4$ units right.

Vertically translated $1$ unit up.

Compare all five graphs. Use the observations you made previously to write a general statement that describes the roles of $a$ , $h$ , and $k$ in the equation $y = a (x - h)^{2} + k$ .

When a is negative, it is reflected about the $x$ -axis.

For $a > 1$ or $a < - 1$ , the parabola is stretched vertically by the positive a value.

For $0 < a < 1$ and $- 1 < a < 0$ , the parabola is compressed vertically by the positive a value.

For $h > 0$ , the parabola is translated horizontally $h$ units right.

For $h < 0$ , the parabola is translated horizontally $h$ units left.

For $k > 0$ , the parabola is translated vertically $k$ units up.

For $k < 0$ , the parabola is translated vertically $k$ units down.

Now it is time to consolidate your skills and knowledge.

Reflection on the function $y = f (x)$ :

Reflection	Mathematical form	Effect
Vertical	$y = - f (x)$	Compared to $y = f (x)$ , the graph of $y = - f (x)$ is a vertical reflection across the x-axis. The point (x,y) on $y = f (x)$ becomes the point (x,-y) on $y = - f (x)$

Stretches on the function $y = f (x)$ :

Stretch	Mathematical form	Effect
Vertical	$y = a f (x)$	If a > 1, the graph is vertically expanded by a factor of a. If 0 < a < 1, the graph is vertically compressed by a factor of a. The point (x,y) on $y = f (x)$ becomes the point (x,ay) on $y = a f (x)$

Translations on the function $y = f (x)$ :

Translation	Mathematical form	Effect
Horizontal	$y = f (x - h)$	Compared to the graph of $y = f (x)$ , the graph of $y = f (x - h)$ is a horizontal translation of h units. when h > 0 the graph is horizontally translated to the RIGHT h units. when h < 0 the graph is horizontally translated to the LEFT h units. The point (x,y) on $y = f (x)$ becomes the point (x+h,y) on $y = f (x)$
Vertical	$y = f (x) + k$	Compared to the graph of $y = f (x)$ , the graph of $y = f (x) + k$ is a horizontal translation of k units. when h > 0 the graph is vertically translated to the RIGHT k units. when h < 0 the graph is vertically translated to the LEFT k units. The point (x,y) on $y = f (x)$ becomes the point (x,y + k) on $y = f (x) + k$

Math journal

At the end of the course, you will fine-tune 8 entries (two from each unit) from your math journal and submit them as your “Culminating assessment - Math journal” (Opens in new window)

Quadratic transformations summary

Summarize quadratic transformations in your math journal with the findings from the investigations. It may resemble the following examples.

The following section shows the graph of $y = a (x - h)^{2} + k$ compared to the graph of $y = x^{2}$ where the given parameter changes.

Parameter:

$a < - 1$ or $a > 1$

Transformations related to $a$

The graph of $y = x^{2}$ is vertically stretched by the positive a value. The resulting parabola is narrower than $y = x^{2}$ .

It is concave up when a is positive and concave down when $a$ is negative, meaning it was reflected about the $x$ -axis.

Parameter:

$- 1 < a < 1$

Transformations related to $a$

The graph of $y = x^{2}$ is vertically compressed by the positive a value . The resulting parabola is wider than $y = x^{2}$ .

It is concave up when a is positive and concave down when a is negative, meaning it was reflected about the $x$ -axis.

Parameter:

$h > 0$

Transformations related to $h$

The graph of $y = x^{2}$ is translated or shifted horizontally by h units to the right.

Parameter:

$h < 0$

Transformations related to $h$

The graph of $y = x^{2}$ is translated or shifted horizontally by $h$ units to the left.

Parameter:

$k > 0$

Transformations related to $k$

The graph of $y = x^{2}$ is vertically translated up by $k$ units.

Parameter:

$k < 0$

Transformations related to $k$

The graph of $y = x^{2}$ is vertically translated down by $k$ units.

If you want more practice with domain and range, examine an interactive tool by entering “transformations of functions” and “Mark Kreie” into your preferred internet search engine. You are encouraged to search for additional materials to support your understanding.

Once you are comfortable with the success criteria, complete the following quiz to assess your progress.

Try “Diagnosing transformations: the value of a in the equation $y = {a x}^{2}$ .”

Start

Now you are ready to investigate further with the following application called “Investigation of transformations: Value of $a$ and $h$ in $y = a (x - h)^{2}$ .”

Write a short paragraph explaining what you explored and how the h values affect the graph. What happens to the graph when the value increases? Decreases? Submit your answer to your portfolio for feedback.

Start

Connecting $a$ and $h$ in $y = a (x - h)^{2}$

How do the values $a$ and $h$ in $y = {a (x - h)}^{2}$ transform the graph of $y = x^{2}$ ?

$a < - 1$ , the graph is reflected about the $x$ axis and has a stretch by the positive $a$ value.

$- 1 < a < 0$ , the graph is reflected about the axis and has a compression by the positive $a$ value.

$0 < a < 1$ , the graph has a compression by the $a$ value.

$a > 1$ , the graph has a stretch by the positive $a$ value.

$h < 0$ , the graph is translated horizontally $h$ units to the left.

$h > 0$ , the graph is translated horizontally $h$ units to the right.

Analyze the following parabola:

$y = 2 {(x + 1)}^{2}$

Analyze the following parabola:

$y = - \frac{7}{9} (x - 5)^{2}$

Analyze the following parabola and sketch its graph.

$y = - 4 (x + 3)^{2}$

Tips for sketching parabola

Determine two symmetrical points.

The following is true for any two symmetrical points on the parabola:

They lie on opposite sides of the axis of symmetry.
They have the same $y$ -value (so they can be connected by a horizontal line segment).
They are the same distance from the axis of symmetry.

Since the axis of symmetry is $x = - 3$ , pairs of symmetrical points can be found using $x$ -values that are the same distance but on opposite sides of $x = - 3$ .

Some possibilities are:

and $x = - 4$ (which produce points that are one unit on either side of $x = - 3$ )
and $x = - 5$ (which produce points that are two units on either side of $x = - 3$ )
and $x = 0$ (which produce points that are three units on either side of $x = - 3$ )

Choose $x = - 2$ and $x = - 4$ . Substitute $x = - 2$ into the equation to find the corresponding $y$ -value. Both points have the same $y$ -value, so two symmetrical points are $(- 2, - 4)$ and $(- 4, - 4)$ .

Use the information you gathered to sketch the graph using pencil and paper.

Analyze the following parabola and sketch its graph.

$y = \frac{2}{3} (x - 1)^{2}$

Describe the transformations that must be applied to the graph of $y = x^{2}$ to get the graph of each equation.

$a = \frac{2}{3}$ , so the graph of $y = x^{2}$ is vertically compressed by $\frac{2}{3}$ .

$h = 1$ , so the graph of $y = x^{2}$ is translated to the right one unit.

State the vertex, the equation of the axis of symmetry, and the concavity.

The vertex is $(1, 0)$ .

The equation of the axis of symmetry is $x = 1$ .

The parabola is concave up.

Determine two symmetrical points.

Since the axis of symmetry is $x = 1$ and $\frac{2}{3}$ has the denominator $3$ , choose values that are three units on either side of $x = 1$ . This will result in an integer $y$ -value that is easier to plot than a fraction.

Choose $x = 4$ and $x = - 2$ .

Substitute $x = 4$ into the equation to find the corresponding $y$ -value.

$y = \frac{2}{3} (4 - 1)^{2}$

$y = \frac{2}{3} (3)^{2}$

$y = \frac{2}{3} (9)$

$y = 6$

Both points have the same $y$ -value, so two symmetrical points are $(4, 6)$ and $(- 2, 6)$ .

Use the information you gathered to sketch the graph using grid paper. To sketch a graph of the function, mark the vertex $(1, 0)$ and the two symmetrical points $(4, 6)$ and $(- 2, 6)$ on a grid. Join the points using a smooth curve to draw a parabola like the one in the following answer.

Write a quadratic function that satisfies these two sets of transformations.

The graph of $y = x^{2}$ is reflected in the $x$ -axis and shifted two units right.

Since the graph is reflected in the $x$ -axis, then $a = - 1$ .

Since the graph is shifted right two units, then $h = 2$ .

Therefore the equation is $f (x) = {- (x - 2)}^{2}$

The vertex of the parabola is $(- 3, 0)$ and the $y$ -intercept is $2$ .

Since the vertex is $(- 3, 0)$ , then $h = - 3$ , so you know the equation thus far is

$y = {a (x + 3)}^{2}$ .

Since the $y$ -intercept is $2$ , then $(0, 2)$ is a point on the parabola.

Use this information to find the value of a in $y = {a (x + 3)}^{2}$ .

Substitute $x = 0$ and $y = 2$ and solve for $a$ .

$2 = a (0 + 3)^{2}$

$2 = a (3)^{2}$

$2 = 9 a$

$\frac{2}{9} = a$

Therefore the equation is $f (x) = \frac{2}{9} (x + 3)^{2}$ .

Now you can test your knowledge and skill with the following application called “Diagnosing transformations in the equation $y = a (x - h)^{2} + k$ .”

Start

Now, you’ll be given another opportunity to test your knowledge of the transformations caused by the values of $h$ and $k$ in the quadratic equation $y = a (x - h)^{2} + k$ .

For each graph, explain its transformation of the parent function $y = x^{2}$ .

The graph of $y = x^{2}$ has been translated to the right $2$ units.

The graph of $y = x^{2}$ has been translated to the left $2$ units.

The graph of $y = x^{2}$ has been translated up $2$ units.

The graph of $y = x^{2}$ has been translated down $2$ units.

The graph of $y = x^{2}$ has been translated to the left $3$ units and down $1$ unit.

The graph of $y = x^{2}$ has been translated to the right $3$ units and down $1$ unit.

The graph of $y = x^{2}$ has been translated to the left $3$ units and up $1$ unit.

The graph of $y = x^{2}$ has been translated to the right $3$ units and up $1$ .

The graph of $y = x^{2}$ has been translated to the right $1$ unit and up $1$ unit.

The graph of $y = x^{2}$ has been translated to the left $1$ unit and down $1$ unit.

Summary of the effect of $h$ and $k$ on the transformation of $y = x^{2}$

Equation	Graph	Summary
$y = {(x - h)}^{2}$ $h > 0$		When $h > 0$ (positive), the graph of $y = {(x - h)}^{2}$ and $y = x^{2}$ are similar because they are the same shape (width) and concave up. They are different because the graph of $y = x^{2}$ is horizontally shifted (translated) to the right $h$ units to get the graph of $y = {(x - h)}^{2}$ . The vertex of the transformed graph is $(h, 0)$ .
$y = {(x - h)}^{2}$ $h < 0$		When $h < 0$ (negative), the equation becomes $y = {(x - (- h))}^{2}$ or $y = {(x + h)}^{2}$ . The graphs of $y = {(x - h)}^{2}$ and $y = x^{2}$ are similar because they are the same shape (width) and concave up. They are different because the graph of $y = x^{2}$ is horizontally shifted (translated) to the left $h$ units to get the graph of $y = {(x + h)}^{2}$ . The vertex of the transformed graph is $(h, 0)$ .
$y = x^{2} + k$ $k > 0$		The graph of $y = x^{2} + k$ , for $k > 0$ , is a vertical translation (or shift) of the graph of $y = x^{2}$ . When $k$ is a positive number, $y = x^{2}$ is translated up $k$ units.
$y = x^{2} + k$ $k < 0$		The graph of $y = x^{2} + k$ , for $k < 0$ , is a vertical translation (or shift) of the graph of $y = x^{2}$ . When $k$ is a negative number, $y = x^{2}$ is translated down $k$ units.
$y = {(x - h)}^{2} + k$		In comparison to the graph of $y = x^{2}$ , the graph of $y = {(x - h)}^{2} + k$ represents a horizontal translation of $h$ units (right when $h$ is positive or left when $h$ is negative), and a vertical translation of $k$ units (either up or down).

Inuit culture and parabolas

Meaning of transformations

Explore this!

Key deas

Need to know

Graphing tool

Graphing quadratic functions

Investigation 1: a>1

y=x2

y=2x2

Notebook

Notebook

Investigation 2: 0<a<1

y=x2 y=12

y=12x2

Notebook

Investigation 3: a<-1

Notebook

Investigation 4: -1<a<0

Notebook

Sketching parabolas of y=a2

Graph properties for y=x2

Graph properties for y=-x2

Symmetrical points

Practice on role of a in y=ax2

Notebook

Investigation 5: h>0

Notebook

Notebook

Investigation 6: h<0

Think!

Notebook

Practice the role of h in y=(x-h)2

Investigation 7: k>0

Notebook

Notebook

Investigation 8: k<0

Think

Notebook

Putting it all together: a, h, and k in y=a(x-h)2+k

Notebook

Notebook

Reflection on the function y = fx:

Stretches on the function y = fx:

Translations on the function y = fx:

Math journal

Connecting a and h in y=a(x-h )2

Tips for sketching parabola

Summary of the effect of h and k on the transformation of y=x2

Investigation 1: $a > 1$

$y = x^{2}$

$y = 2 x^{2}$

Investigation 2: $0 < a < 1$

$y = x^{2}$ $y = \frac{1}{2}$

$y = \frac{1}{2} x^{2}$

Investigation 3: $a < - 1$

Investigation 4: $- 1 < a < 0$

Sketching parabolas of $y = a^{2}$

Graph properties for $y = x^{2}$

Graph properties for $y = {- x}^{2}$

Practice on role of $a$ in $y = a x^{2}$

Investigation 5: $h > 0$

Investigation 6: $h < 0$

Practice the role of $h$ in $y = {(x - h)}^{2}$

Investigation 7: $k > 0$

Investigation 8: $k < 0$

Putting it all together: $a$ , $h$ , and $k$ in $y = a (x - h)^{2} + k$

Reflection on the function $y = f (x)$ :

Stretches on the function $y = f (x)$ :

Translations on the function $y = f (x)$ :

Connecting $a$ and $h$ in $y = a (x - h)^{2}$

Summary of the effect of $h$ and $k$ on the transformation of $y = x^{2}$