Alternative Format: Lesson 1 — Transformations of \(y = x^2\)

Let's Start Thinking

Transformations​

If you have ever sat lakeside on a calm day, you've likely seen the reflection of a shoreline in the surface of the water.

A photo of tree reflected in a lake.

Source: Forest - GlowingEarth/iStock/Getty Images

Perhaps you've received a photocopied paper from someone where the copy you receive is either larger or smaller than the original image?

A photo of a photocopier creating copies.

Source: Photocopier - GoodLifeStudio/iStock/Getty Images

Both of these familiar scenarios are governed by rules that are applied to an object to produce an image that has been changed in some way. In mathematics, these rules are called transformations.

In this lesson, we will investigate various transformations.


Lesson Goals

  • Determine the image of a set of points under a translation, reflection, or stretch (or compression).
  • Describe the role of \(a\) in \(y=ax^2\).
  • Describe the role of \(k\) in \(y=x^2+k\).
  • Describe the role of \(h\) in \(y=(x-h)^2\).

Translations, Reflections, and Vertical Stretches


Introduction to Transformations

A transformation is a rule that can be applied to an object to change its location, shape, size, or orientation.

We have names for both the initial object and the changed object.

  • Pre-image: the initial object
  • Image: the resulting object after the transformation is applied
  • \(A'\): the image of \(A\)

Did You Know?

\(A'\) is read “A prime.”

Consider \(\triangle ABD\), as shown.

Triangle ABC, where A is at point (0, 3), B is at point (4, 0), and D is at point (negative 1, negative 3).

Let's transform \(\triangle ABD\) by sliding it \(2\) units to the left and \(1\) unit down.

Triangle A prime B prime C prime, where A prime is at point (negative 2, 2), B prime is at point (2, negative 1), and D prime is at point (negative 3, negative 4).

In general, a transformation can change an object's location, shape, size, or orientation.

In this particular transformation, we have changed the triangle's location, but not its shape or size. 

As for orientation, notice that in \(\triangle ABD\), when we move from \(A\) to \(B\) to \(D\), we would say that it has a clockwise orientation. Likewise, \(\triangle A'B'D'\) also has a clockwise orientation.

The orientation of both ABD and A prime B prime D prime is clockwise.

Thus, orientation is unchanged.

Summary of Translations

  • Location: changed
  • Shape: unchanged
  • Size: unchanged
  • Orientation: unchanged

Types of Transformations

The three types of transformations that we are going to investigate in this lesson are:

  • Translations, where we shift the object to a different location.

    A square is shifted 2 units right, 3 units down.

  • Reflections, where we flip the object over a line of reflection.

    A trapezoid is reflected over a line.

  • Stretches, where we lengthen the object's distance from a fixed line using a stretch factor.

    A triangle is stretched from a fixed line.

What Is a Translation?

Let's start with translations.

A translation is a transformation in which every point of an object moves in the same direction by the same distance.

A translation must always be specified by both a direction and a distance.

For example, here, \(\triangle ABC\) is translated or shifted \(14\) units right and \(3\) units up to form \(\triangle A'B'C'\).

Check Your Understanding 1

Question

Describe the translation for \(\triangle ABC\) using the number of units and the direction that \(\triangle ABC\) has be translated.

Triangle A prime B prime C prime lies 8 units to the right and 8 units down from triangle ABC.

Created with GeoGebra. Author: University of Waterloo. CC BY-NC-SA 4.0. https://ggbm.at/gpbaszj6

Answer

\(\triangle ABC\) has been translated \(10\) units right and \(5\) units down to form \(\triangle A'B'C'\).

Feedback

The use arrows can help you to find the correct translations.

Arrows from triangle ABC to triangle A prime B prime C prime point 8 units right and a second arrow points 8 units down.

Interactive Version

Translations

What Is a Reflection?

Next, let's look at reflections.

A reflection is a transformation that "flips" an object over a line of reflection (also called a mirror line) such that the line of reflection is the perpendicular bisector of each pre-image point and its image point.

Let's consider an example.

Quadrilateral \(PQRS\) is mapped onto \(P'Q'R'S'\), by being reflected in the dotted line.

Quadrilateral PQRS is reflected across a line to form quadrilateral P prime Q prime R prime S prime. Additionally the line from a point in PQRS and it corresponding point in P prime Q prime R prime S prime forms a right angle with the line of reflection.

Consider \(Q'\), its exact position can be verified by forming line segment \(QQ'\) and ensuring that \(QQ'\) is perpendicular to the mirror line and that \(Q\) and \(Q'\) are the same perpendicular distance to the mirror line.

Every point in PQRS is equidistant from the reflection line as its corresponding point in P'Q'R'S'. The line connecting point Q and point Q prime measures 5 units from point Q to the line of reflection and 5 units from point Q prime to the line of reflection.

Notice that under a reflection, an object's shape and size remain unchanged.

The pre-image and the image are congruent. But the object's location and orientation may change.

In terms of orientation, \(PQRS\) is labelled clockwise. Whereas the image, \(P'Q'R'S'\), is labelled counterclockwise.

Summary of Reflections

  • Shape: unchanged
  • Size: unchanged
  • Location: changed
  • Orientation: changed

Check Your Understanding 2

Question

\(\triangle ABC\) has vertices \(A(4,-2)\), \(B(-2,-7)\), and \(C(5,-6)\).

What are the vertices of its reflection in the \(x\)-axis?

Answer

\(\triangle A'B'C'\) has vertices \(A'(4,2)\), \(B'(-2,7)\), and \(C'(5,6)\).

Created with GeoGebra. Author: University of Waterloo. CC BY-NC-SA 4.0.

Feedback

Remember that in a reflection, each point is the same distance from the reflection line as its image.

Interactive Version

Reflections

What Is a Stretch?

Finally, let's consider stretches.

A stretch is a transformation that increases the perpendicular distance of each point of an object from a fixed line by a given factor.

So here we have \(\triangle ABC\) and it's mapped onto \(\triangle A'B'C'\) by being vertically stretched by a factor of \(2\) about the \(x\)-axis.

In triangle ABC, point A lies at (1, 0), point B lies at (negative 2, 4), and point C lies at (3, 2). In triangle A prime B prime C prime, point A prime lies at (1, 0), point B prime lies at (negative 2, 8), and point C prime lies at (3, 4).

Take a look at point \(B(-2, 4)\). Notice that \(B\) is \(4\) units from the \(x\)-axis.

This perpendicular distance is stretched by a factor of \(2\) so that \(B'\) must be at the point \((-2, 8)\).

Point B prime lies twice the distance from the x-axis as point B.

Similarly, point \(C(3,2)\) is \(2\) units from the \(x\)-axis.

So \(C'\) must be \(2 \times 2 = 4\) units from the \(x\)-axis.

Point C prime lies twice the distance from the x-axis as point C.

And since \(A\) is right on the \(x\)-axis, \(A'\) remains unchanged.

Check Your Understanding 3

Question

Find the image of \(\triangle ABC\) under a vertical stretch by a factor of \(3\).

In triangle ABC point A lies at (2, negative 4), point B lies at (negative 2, 2), and point C lies at (5, 0).

Answer

The vertices of \(\triangle A'B'C'\) are located at

  • \(A'(2,-12)\)
  • \(B'(-2, 6)\)
  • \(C'(5,0)\)

Each of the vertices of A prime B prime C prime lie 3 times the distance as the vertices of ABC from the x-axis.

Created with GeoGebra. Author: University of Waterloo. CC BY-NC-SA 4.0.

Feedback

Remember that in a vertical strech, the \(x\)-coordinate of each point remains the same, and the \(y\)-coordinate is multiplied by the stretch factor.

Interactive Version

Graphing Vertical Stretches

Transformations

Transformations are powerful tools used to manipulate graphs or images.

Math in Action

Transformations are used frequently in computer graphics. For instance, a video game character can be made to turn around and move in the opposite direction using a combination of translations and a reflection.


Transformations as Mappings

The transformations discussed previously can also be described as a mapping of a pre-image point to its image point.

Translations

Recall

A translation is a transformation in which every point of an object moves in the same direction by the same distance.

For any non-zero real number \(h\), a horizontal translation of \(h\) units right (left if \(h\) is negative) maps \(A(x,y)\) onto \(A'(x+h,y)\).

For any non-zero real number \(k\), a vertical translation of \(k\) units up (down if \(k\) is negative) maps \(A(x,y)\) onto \(A'(x,y+k)\).

Reflections

Recall

A reflection is a transformation that 'flips' an object over a line of reflection (also called a mirror line) such that the line of reflection is the perpendicular bisector of each pre-image point and its image point.

A reflection in the \(x\)-axis maps \(A(x,y)\) onto \(A'(x,-y)\).

Reflections in the \(y\)-axis will be considered in a different lesson.

Stretches

Recall

A stretch is a transformation that increases the perpendicular distance of each point of an object from a fixed line by a given factor.

For any positive real number \(k\), a vertical stretch by a factor of \(k\) maps \(A(x,y)\) onto \(A'(x,ky)\).

Notice that

  • the \(x\)-coordinate remains the same, and
  • the \(y\)-coordinate is multiplied by the factor \(k\).

A vertical compression results when \(0 \lt k \lt1\).

Horizontal reflections and compressions will be considered in a different lesson.


Vertical Stretches and Reflections of Parabolas


Explore This 1

Question

Looking at the graphs below, what is the role of \(a\) in \(y = ax^2\)?

\(a = 3\)

The function y equals 3 times x squared is vertically stretched by a factor of 3 when compared to the original function y equals x squared.

\(a = 2\)

The function y equals 2 times x squared is vertically stretched by a factor of 2 when compared to the original function y equals x squared.

\(a = \dfrac{1}{2}\)

The function y equals 1 over 2 times x squared is vertically compressed by a factor of 2 when compared to the original function y equals x squared.

\(a = 0\)

The function y equals 0 is a horizontal line along the x-axis.

\(a = -\dfrac{1}{2}\)

The function y equals negative 1 over 2 times x squared is reflected across the x-axis and vertically compressed by a factor of 2 when compared to the original function y equals x squared.

\(a = -1\)

The function y equals negative 1 times x squared is reflected across the x-axis when compared to the original function y equals x squared.

\(a = -2\)

The function y equals negative 2 times x squared is reflected across the x-axis and vertically stretched by a factor of 2 when compared to the original function y equals x squared.

\(a = -3\)

The function y equals negative 3 times x squared is reflected across the x-axis and vertically stretched by a factor of 3 when compared to the original function y equals x squared.

Created with GeoGebra. Author: University of Waterloo. CC BY-NC-SA 4.0.

Interactive Version

Transformation of Quadratics


Explore This 1 Summary

In Explore This 1, you may have noticed the following:

  • When \(a\) is negative (\(a \lt 0\)), \(y=ax^2\) opens down.
  • When the absolute value of \(a\) is greater than \(1\), there is a vertical stretch.
  • When the absolute value of \(a\) is less than \(1\), there is a vertical compression.

The absolute value of a number is the number's distance from \(0\); it is the number with the sign ignored.

For example, the absolute value of \(3\) is \(3\); the absolute value of \(-1.4\) is \(1.4\).

Transforming \(y=x^2\) onto \(y=ax^2\)

The graph of \(y=x^2\) is mapped onto \(y=ax^2\) in the following ways.

\(a\) Value Reflection Stretch/Compression Graph

\(a \gt 1\)

 None

Vertical stretch

\(a=1\)

 None None

\(0 \lt a \lt 1\)

 None

Vertical compression

\(-1 \lt a \lt 0\)

 Reflection in the \(x\)-axis

Vertical compression

\(a=-1\)

 Reflection in the \(x\)-axis None

\(a \lt -1\)

Reflection in the \(x\)-axis

Vertical stretch

Describing Vertical Compressions

When applied to a point, a vertical compression shortens the point's distance from the \(x\)-axis by a compression factor. For instance, to graph \(y=\dfrac{1}{3}x^2\) each point of \(y=x^2\) is compressed toward the \(x\)-axis by

  • multiplying the point's \(y\)-coordinate by \(\dfrac{1}{3}\), or equivalently
  • dividing the point's \(y\)-coordinate by \(3\).

Therefore, the transformation that maps \(y=x^2\) onto \(y=\dfrac{1}{3}x^2\) can be described as

  • a vertical stretch by a factor of \(\dfrac{1}{3}\), or equivalently
  • a vertical compression by a factor of \(3\).

Example 1

State the transformations that map \(y=x^2\) onto \(y=-\dfrac{1}{2}x^2\) and draw both graphs.

 Solution

  • Since \(a\) is negative, there is a reflection in the \(x\)-axis.
  • Since the absolute value of \(a\) is \(\dfrac{1}{2}\) there is a vertical compression by a factor of \(2\); or, equivalently, this can be called a vertical stretch by a factor of \(\dfrac12\).

To draw the graph of \(y=-\dfrac{1}{2}x^2\), we first graph \(y=x^2\).

\(x\) \(y=x^2\)
\(-3\) \(9\)
\(-2\) \(4\)
\(-1\) \(1\)
\(0\) \(0\)
\(1\) \(1\)
\(2\) \(4\)
\(3\) \(9\)
The graph of y equals x squared with the points (negative 3, 9), (0,0), and (2,4) highlighted.

For each point on the graph:

  • Apply two transformations: reflection in the \(x\)-axis and vertical compression by a factor of \(2\).
  • This is achieved in one step by multiplying all \(y\)-values by \(\dfrac{-1}{2}\), the value of \(a\). Remember to keep the \(x\)-values the same.

For example,

  • \((-3,9)\) maps onto \(\left(-3,\dfrac{-9}{2}\right)\),
  • \((0,0)\) maps onto \(\left(0,0 \right)\), and
  • \((2,4)\) maps onto \((2,-2)\).

Doing the same for each point on the graph of \(y=x^2\), we get:

The graph y equals negative one half times x squared with the points (negative 3, negative 9 over 2), (0, 0), and (2, negative 2) highlighted.


Check Your Understanding 4

Question

Consider the transformation that maps \(y = x^2\) onto \(y = -2x^2\).

  1. Determine if there is a reflection in the \(x\)-axis.
  2. Determine if a vertical stretch or vertical compression is used and by what factor it is applied.
  3. Map the points  \((-2, 4)\), \((-1,1)\), \((0,0)\), \((1,1)\), and \((2, 4)\) on the original function, \(y = x^2\), to their corresponding points on \(y = -2x^2\).

The function y equals x squared passes through the points (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), and (2, 4).

Answer

  1. There is a reflection in the \(x\)-axis.
  2. There is a vertical stretch by a factor of \(2\).
  3. The points \((-2, 4)\), \((-1,1)\), \((0,0)\), \((1,1)\), and \((2, 4)\) on the original function, \(y = x^2\), map to  \((-2, -8)\), \((-1,-2)\), \((0,0)\), \((1,-2)\), and \((2, -8)\) and the transformed function, \(y = -2x^2\).

The function y equals negative 2 times x squared passes through the points (negative 2, negative 8), (negative 1, negative 2), (0, 0), (1, negative 2), and (2, negative 8).

Created with GeoGebra. Author: University of Waterloo. CC BY-NC-SA 4.0.

Feedback

In this example, \(a = -2\).

Since \(a \lt 0\), there is a reflection in the \(x\)-axis.

Since \(a \lt -1\), there is a vertical stretch by a factor of \(2\).

To find the points on the transformed function, multiply the \(y\)-coordinates of the points on \(y = x^2\) by \(-2\) and leave the \(x\)-values the same.

Interactive Version

Function Transformations


Vertical and Horizontal Translations of Parabolas


Explore This 2

Question

Looking at the graphs below, what is the role of \(k\) in \(y=x^{2}+k\)?

\(k = 10\)

The parabola has been translated up so the vertex is at the point (0, 10) rather than the point (0, 0).

\(k = 4\)

The parabola has been translated up so the vertex is at the point (0, 4) rather than the point (0, 0).

\(k = -3\)

The parabola has been translated down so the vertex is at the point (0, negative 3) rather than the point (0, 0).

\(k = -8\)

The parabola has been translated down so the vertex is at the point (0, negative 8) rather than the point (0, 0).

Created with GeoGebra. Author: University of Waterloo. CC BY-NC-SA 4.0.

Interactive Version

Vertical Translations


Explore This 2 Summary

In Explore This 2, you may have noticed the following.

  • When \(k\) is negative (\(k \lt 0\)), the graph shifts down from \(y=x^2\).
  • When \(k\) is positive (\(k \gt 0\)), the graph shifts up from \(y=x^2\).

Transforming \(y=x^2\) onto \(y=x^2+k\)

The graph of \(y=x^2\) is mapped onto \(y=x^2+k\) in the following ways.

\(k\) value Translation

\(k \lt 0\)

e.g., \(y=x^2-4\)

Translate down

e.g., Translate 4 units down

\(k \gt 0\)

e.g., \(y=x^2+3\)

Translate up

e.g., Translate 3 units up

Example 2

State the transformation that maps \(y=x^2\) onto \(y=x^2-3\) and draw both graphs.

Solution

  • Since \(k\) is negative, the graph moves down. The transformation is a translation \(3\) units down.

To draw the graph of \(y=x^2-3\), we first graph \(y=x^2\).

\(x\) \(y=x^2\)
\(-3\) \(9\)
\(-2\) \(4\)
\(-1\) \(1\)
\(0\) \(0\)
\(1\) \(1\)
\(2\) \(4\)
\(3\) \(9\)
The graph y equals x squared with the point (negative 3, 9) highlighted.

 

For each point on the graph:

  • Apply the transformation: translate \(3\) units down.
  • This is achieved by subtracting \(3\) from each \(y\)-value. Remember to keep the \(x\)-values the same.

For example,

  • \((-3,9)\) maps onto \(\left(-3,6\right)\),
  • \((0,0)\) maps onto (\(0,-3)\), and
  • \((2,4)\) maps onto \((2,1)\).

Doing the same for each point on the graph of \(y=x^2\), we get:

The graph y equals x squared minus 3 with the point (negative 3, 6) highlighted.


Check Your Understanding 5

Question

Consider the transformation that maps \(y = x^2\) onto \(y = x^2 - 3\).

  1. State the direction of the translation and.
  2. Map the points \((-2, 4)\), \((-1,1)\), \((0,0)\), \((1,1)\), and \((2, 4)\) on the original function, \(y = x^2\), to their corresponding points on \(y = x^2 -3\).

The function y equals x squared passes through the points (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), and (2, 4).

Answer

  1. There is a translation of \(3\) units down.
  2. The points \((-2, 4)\), \((-1,1)\), \((0,0)\), \((1,1)\), and \((2, 4)\) on the original function, \(y = x^2\), map to  \((-2, 1)\), \((-1,-2)\), \((0,-3)\), \((1,-2)\), and \((2, 1)\) and the transformed function, \(y = x^2 -1\).

The function y equals x squared minus 3 passes through the points (negative 2, 1), (negative 1, negative 2), (0, negative 3), (1, negative 2), and (2, 1).

Created with GeoGebra. Author: University of Waterloo. CC BY-NC-SA 4.0.

Feedback

When mapping \(y=x^{2}\) onto \(y=x^{2}+k\), there is a translation up whenever \(k \gt 0\), and a translation down whenever \(k \lt 0\).

There is a translation of \(3\) units down.

Interactive Version

Vertical Translations


Explore This 3

Question

Looking at the graphs below, what is the role of \(h\) in \(y= (x- h)^{2}\)?

\(h = -8\)

The parabola y equals x squared has been translated left so the vertex is at the point (0, negative 8) rather than the point (0, 0), this transformed parabola is y equals open bracket x plus 8 close bracket squared.

\(h = -3\)

The parabola y equals x squared has been translated left so the vertex is at the point (0, negative 3) rather than the point (0, 0), this transformed parabola is y equals open bracket x plus 3 close bracket squared.

\(h = 4\)

The parabola y equals x squared has been translated right so the vertex is at the point (0, 4) rather than the point (0, 0), this transformed parabola is y equals open bracket x minus 3 close bracket squared.

\(h = 7\)

The parabola y equals x squared has been translated right so the vertex is at the point (0, 7) rather than the point (0, 0), this transformed parabola is y equals open bracket x minus 7 close bracket squared.

Created with GeoGebra. Author: University of Waterloo. CC BY-NC-SA 4.0.

Interactive Version

Horizontal Translations


Explore This 3 Summary

In Explore This 3, you may have noticed the following.

  • When \(h\) is negative (\(h \lt 0\)),
    • the graph shifts left from \(y=x^2\), and
    • the equation has a "\(+\)" inside the bracket.
  • When \(h\) is positive (\(h \gt 0\)),
    • the graph shifts right from \(y=x^2\), and
    • the equation has a "\(-\)" inside the bracket.

Transforming \(y=x^2\) onto \(y=(x-h)^2\)

The graph of \(y=x^2\) is mapped onto \(y=(x-h)^2\) in the following ways.

\(h\) value Translation

\(h \lt 0\)

e.g., \(y=(x+4)^2\)

Translate left

e.g., Translate 4 units left

\(h \gt 0\)

e.g., \(y=(x-3)^2\)

Translate right

e.g., Translate 3 units right

Example 3

State the transformation that maps \(y=x^2\) onto \(y=(x-3)^2\) and draw both graphs.

 Solution

  • The sign of \(h\) is opposite of the sign in the bracket.
    • Since the bracket has a subtraction, \(h\) is positive.
  • Since \(h\) is positive, the graph moves right. The transformation is a translation \(3\) units right.

To draw the graph of \(y=(x-3)^2\), we first graph \(y=x^2\).

\(x\) \(y=x^2\)
\(-3\) \(9\)
\(-2\) \(4\)
\(-1\) \(1\)
\(0\) \(0\)
\(1\) \(1\)
\(2\) \(4\)
\(3\) \(9\)
The graph y equals x squared with the point (-3, 9) highlighted.

For each point on the graph:

  • Apply the transformation: translate \(3\) units right.
  • This is achieved by adding \(3\) to each \(x\)-value. Remember to keep the \(y\)-values the same.

For example,

  • \((-3,9)\) maps onto \(\left(0,9\right)\),
  • \((0,0)\) maps onto \((3,0)\), and
  • \((2,4)\) maps onto \((5,4)\).

Doing the same for each point on the graph of \(y=x^2\), we get:

The graph y equals open bracket x minus 3 close bracket squared with the point (0, 9) highlighted.


Check Your Understanding 6

Question

State the transformation that maps \(y=x^{2}\) onto \(y=(x+3)^{2}\) by transforming the points on the original function, \(y = x^2\).

The function y equals x squared passes through the points (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), and (2, 4).

Answer

The function y equals open brackets x plus 3 close brackets squared passes through the points (negative 5, 4), (negative 4, 1), (negative 3, 0), (negative 2, 1), and (negative 1, 4).

Created with GeoGebra. Author: University of Waterloo. CC BY-NC-SA 4.0.

There is a translation of \(3\) units left.

Feedback

When mapping \(y=x^{2}\) onto \(y=(x-h)^{2}\), the sign of \(h\) is opposite of the sign in the bracket.

In this case, since the bracket has a subtraction sign, \(h\) is positive.

There is a translation right whenever \(h>0\) and a translation left whenever \(h>0\).

The size of the translation is given by the size of \(h .\)

In this case, since \(h=-3\), there is a translation of \(3\) units left.

Interactive Version

Horizontal Translations


Wrap-Up


Lesson Summary

  • The graph of \(y=ax^2\) is a transformation of the graph of \(y=x^2\).
    • The transformation is a vertical stretch (or compression) by a factor determined from the size of \(a\).
    • There is a reflection in the \(x\)-axis if \(a\) is negative.
  • The graph of \(y=x^2+k\) is a transformation of the graph of \(y=x^2\).
    • The transformation is a vertical translation (up or down) given by the size of \(k\).
  • The graph of \(y=(x-h)^2\) is a transformation of the graph of \(y=x^2\).
    • The transformation is a horizontal translation (right or left) given by the size of \(h\).
    • Be careful with the sign of \(h\)! It is opposite of the sign inside the bracket. For example, if there is a subtraction inside the simplified bracket, then \(h\) is positive and the translation is \(h\) units right.

Take It With You

In this lesson, we compared \(y=x^2\) and each of

  • \(y=ax^2\)
  • \(y=(x-h)^2\)  
  • \(y=x^2+k\)

It is possible to combine our findings together and compare \(y=a(x-h)^2+k\) to \(y=x^2\).

See if you can state the transformations that will map \(y=x^2\) onto \(y=-2(x+3)^2-1\) and graph the image parabola.