Graph Transformations - Math Steps, Examples & Questions

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Graph transformations

Here you will learn about graph transformations, including translating vertically, translating horizontally, and reflecting in the coordinate axes.

Students will first learn about graph transformations as part of geometry in $8$ th grade.

What are graph transformations?

Graph transformations involve performing transformations such as translations and reflections on the graph of a function.

You may be asked to sketch a graph after a given transformation or asked to write down the position of a coordinate after a transformation has been applied.

To do this, you need to understand what each of the graph transformations look like and how they relate to the original function (also called the parent function).

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Transformations of functions

Let’s use a simple function such as $y=x^2$ to illustrate translations.

First you can write the given function using function notation and draw the graph using a table of values to help.

Translating the graph in a vertical direction

This can be done by adding or subtracting a constant from the $y$ -coordinate. The transformed graph is blue.

So translating vertically by the vector $\left(\begin{matrix}0 \\ a \end{matrix}\right)$ can be done using the transformation $f(x)+a.$

Translating the graph in a horizontal direction

This can be done by adding or subtracting a constant from the $x$ -coordinate. The transformed graph is blue.

So translating vertically by the vector $\left(\begin{matrix}a \\ 0 \end{matrix}\right)$ can be done using the transformation $f(x-a).$

Notice how the transformation $f(x+1)$ translated the graph to the left and not the right.

Reflecting a graph of a function

The function $y=f(x)$ has a point $(1, \, 3)$ as shown.

You will need to be able to apply all of these transformations to coordinates marked on unknown functions as well as sketch transformations of known functions such as the graphs of $\sin(x), \, \cos(x)$ and $\tan(x).$

Combinations of transformations

The different translations and reflections can be combined.

For example,

Further study

In higher level mathematics, these transformations of functions are looked at in more depth to include a horizontal stretch $f(ax)$ and a vertical stretch $af(x).$

These, and more complicated transformations, are applied to functions such as polynomials, exponentials, inverse functions and more complicated trigonometric functions.

What are graph transformations?

Common Core State Standards

How does this relate to $8$ th grade math and high school math?

Grade 8 – Geometry (8.G.A.3)
Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

High School – Geometry – Congruence (HS.G.CO.A.2)
Represent transformations in the plane using, for example, transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (example, translation versus horizontal stretch).

How to use graph transformations

In order to use graph transformations:

Determine whether the transformation is a translation or reflection.
Choose the correct transformation to apply.
Write down the required coordinate or sketch the graph.

Graph transformations examples

Example 1: applying a translation in the y-direction

The diagram shows the graph of $y=f(x)$ and a point on the graph $P(2, \, 5).$

State the coordinate of the image of point $P$ on the graph $y=f(x)- \, 4.$

Determine whether the transformation is a translation or reflection.

The function has been translated.

2Choose the correct transformation to apply.

$f(x)+a$ is a translation in the $y$ -direction.

$f(x)- \, 4$ is a translation by the vector $\left(\begin{matrix} \, \, \, \, 0 \\ - \, 4 \end{matrix}\right).$ You need to subtract $4$ from the $y$ -coordinate.

3Write down the required coordinate or sketch the graph.

5-4=1

Therefore, the image of coordinate $P$ will be $(2, \, 1).$

Example 2: applying a translation in the x-direction

The diagram shows the graph of $y=f(x)$ and a point on the graph $P(2, \, 5).$

State the coordinate of the image of point $P$ on the graph $y=f(x+5).$

Determine whether the transformation is a translation or reflection.

The function has been translated.

Choose the correct transformation to apply from the rules:

$f(x-a)$ is a translation in the $x$ -direction.

$f(x+5)$ is a translation by the vector $\left(\begin{matrix}- \, 5 \\ \, \, \, \, 0 \end{matrix}\right).$ You need to subtract $5$ from the $x$ -coordinate.

Write down the required coordinate or sketch the graph.

2-5=- \, 3

Therefore, the image of coordinate $P$ will be $(- \, 3, \, 5).$

Example 3: applying a reflection in the x-axis

The diagram shows the graph of $y=f(x)$ and a point on the graph $P(2, \, 5).$

Sketch the graph and state the coordinate of the image of point $P$ on the graph $y=- \, f(x).$

Determine whether the transformation is a translation or reflection.

The function has been reflected.

Choose the correct transformation to apply.

$- \, f(x)$ is a reflection in the $x$ -axis.

You need to multiply the $y$ -coordinates by $- \, 1.$

Write down the required coordinate or sketch the graph.

The image of the coordinate of $P$ will be $(2, \, - \, 5).$

Example 4: applying a reflection in the y-axis

The diagram shows the graph of $y=f(x)$ and a point on the graph $P(2, \, - \, 5).$

Sketch the graph and state the coordinate of the image of point $P$ on the graph $y=f(- \, x).$

Determine whether the transformation is a translation or reflection.

The function has been reflected.

Choose the correct transformation to apply from the rules:

$f(- \, x)$ is a reflection in the $y$ -axis.

You need to multiply the $x$ -coordinates by $- \, 1.$

Write down the required coordinate or sketch the graph.

The image of coordinate $P$ will be $(- \, 2, \, - \, 5).$

Example 5: applying a combination of translations

The image shows the graph of the quadratic function $f(x)$ which has a turning point at $(- \, 3, \, - \, 2).$

Sketch the graph of the function $f(x-4)+3,$ labeling the coordinate of the turning point.

Determine whether the transformation is a translation or reflection.

The function has been translated in both directions.

Choose the correct transformation to apply from the rules:

$f(x)+a$ is a translation in the $y$ -direction.

$f(x-a)$ is a translation in the $x$ -direction.

$f(x-4)+3$ is a translation by the vector $\left(\begin{matrix}4 \\ 3 \end{matrix}\right).$ You need to add $4$ to the $x$ -coordinate and add $3$ to the $y$ -coordinate.

Write down the required coordinate or sketch the graph.

- \, 3+4=1

$- \, 2+3=1$

Therefore, the turning point will be $(1, \, 1).$

Example 6: applying a combination of reflections and translations

The image shows the graph of the quadratic function $f(x)$ which has a turning point at $(- \, 3, \, - \, 2).$

Sketch the graph of the function $f(- \, x)- \, 1,$ labeling the coordinate of the turning point.

Determine whether the transformation is a translation or reflection.

The function has been reflected and translated.

Choose the correct transformation to apply from the rules:

$f(x)+a$ is a translation in the $y$ -direction.

$f(- \, x)$ is a reflection in the $y$ -axis.

$f(- \, x)- \, 1$ is a reflection in the $y$ -axis followed by a translation by the vector $\left(\begin{matrix} \, \, \, \, 0 \\ - \, 1 \end{matrix}\right).$

You need to multiply the $x$ -coordinate by $- \, 1$ and then subtract $1$ from the $y$ -coordinate.

Write down the required coordinate or sketch the graph

\begin{aligned}&- \, 3\times{- \, 1}=3 \\\\ &- \, 2-1=- \, 3 \end{aligned}

Therefore, the turning point will be $(3, \, - \, 3).$

Example 7: identifying a transformation

The curve with equation $y=f(x)$ is translated so that the minimum point at $(- \, 1, \, 0)$ is translated to $(5, \, 0).$ Find the equation of the new curve.

Determine whether the transformation is a translation or reflection.

The function has been translated.

Choose the correct transformation to apply.

$f(x-a)$ is a translation in the $x$ -direction.

Write down the required coordinate or sketch the graph.

In this case, you need to write the equation of the new graph.

The graph has been translated by the vector $\left(\begin{matrix}6 \\ 0 \end{matrix}\right).$

The equation of the new graph is $y=f(x-6).$

Teaching tips for graph transformations

Cover one type of transformation at a time: vertical translations, horizontal translations, reflections, and stretches/compressions. This approach helps students grasp each concept individually before combining them.

Begin with common basic functions (for example, linear, quadratic, and absolute value) to introduce students to the foundational shapes of graphs. Show how each type of transformation (shift, stretch, reflection) alters these basic functions.

Use graphing software to demonstrate transformations in real-time. Showing a live graph helps students see exactly how shifts, stretches, and reflections affect the shape of the graph.

Encourage students to identify and label key points, such as the vertex of a parabola or the $y$ -intercept, on the original graph. Then, have them track how these points shift in the new function after each transformation, which helps maintain accuracy in their sketches.

Easy mistakes to make

Performing the wrong horizontal translation
A common mistake is thinking the transformation of $f(x+2)$ will mean the function translates to the right by $2.$ You can see that if you apply the transformation to the function, find a table of values and then plot the points, the function actually translates to the left.

This is because, when you use the transformation $f(x+2),$ the $y$ -values you obtain are for values of $x$ that would normally be two places to the right, so you are shifting those points to the left.

$f(x+2)$ moves the graph to the left by $2, \, f(x-2)$ moves the graph to the right by $2.$

Reflecting in the wrong axes
A common mistake is to confuse the following functions: $- \, f(x)$ and $f(- \, x).$

If $y=f(x), \, - \, f(x)=- \, y.$ This means the $y$ -values are being multiplied by $- \, 1.$

Applying $f(- \, x)$ is changing the sign of the $x$ -value before applying the function.

It’s the same as taking the coordinate axes and switching the signs on the numbers on the $x$ -axis.

Independent and dependent variables
Interpreting graphs

Practice graph transformations questions

(7, \, 1)

(- \, 1, \, 1)

(3, \, – \, 3)

(3, \, 5)

$f(x-4)$ is a translation by $\left(\begin{matrix}4 \\ 0 \end{matrix}\right).$ Add $4$ to the $x$ -coordinate.

(- \, 2, \, 1)

(- \, 2, \, 5)

(0, \, 3)

(- \, 4, \, 3)

$f(x)+2$ is a translation by $\left(\begin{matrix}0 \\ 2 \end{matrix}\right).$ Add $2$ to the $y$ -coordinate.

(5, \, 1)

(- \, 1, \, 5)

(1, \, – \, 5)

(- \, 5, \, – \, 1)

$f(- \, x)$ is a reflection in the $y$ -axis. Multiply the $x$ -coordinate by $– \, 1.$

(4, \, 2)

(- \, 4, \, 2)

(4, \, – \, 2)

(- \, 2, \, – \, 4)

$– \, f(x)$ is a reflection in the $x$ -axis. Multiply the $y$ -coordinate by $– \, 1.$

(5, \, – \, 4)

(5, \, 6)

(- \, 1, \, 6)

(- \, 1, \, – \, 4)

$f(x+3)+5$ is a translation by $\left(\begin{matrix}- \, 3 \\ \, \, \, \, 5 \end{matrix}\right).$ Subtract $3$ from the $x$ -coordinate and add $5$ to the $y$ -coordinate.

(- \, 5, \, – \, 5)

(- \, 5, \, 1)

(5, \, – \, 1)

(5, \, 5)

$– \, f(x)-3$ is a reflection in the $x$ -axis followed by a translation by $\left(\begin{matrix} \, \, \, \, 0 \\ – \, 3 \end{matrix}\right).$ Multiply the $y$ -coordinate by $– \, 1,$ then subtract $3.$

Graph transformations FAQs

What are the basic types of graph transformations?

The main types of graph transformations include horizontal shifts (left or right), vertical shifts (up or down), reflections (flipping over the $x$ -axis or $y$ -axis), and stretches/compressions, also called dilations, (making the graph wider, narrower, taller, or shorter).

How can I identify transformations from a function’s equation?

Look at the placement of constants in relation to $f(x)\text{:}$

Outside the function $($ like $f(x)+k$ or $af(x))$ affects vertical shifts or stretches.

Inside the function $($ like $f(x+h)$ or $f(bx))$ affects horizontal shifts or stretches.

How do graph transformations relate to inequalities?

When dealing with inequalities $($ example, $f(x)\geq{0})$ graph transformations can change the regions where the inequality holds true.

For example, shifting or reflecting the graph affects where the graph is above or below the $x$ -axis.

The next lessons are

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