# Graph Translations and Reflections

## Graph Translations and Reflections Revision

**Graph Translations and Reflections**

We shall look at two types of **graph transformations**.

Firstly we shall look at **translations**, which move the position of the graph up, down, left and right.

Then we shall look at **reflections**, where the graph is reflected over a line of reflection.

**Horizontal Translations**

**Horizontal translations** move the graph left and right. This is along the** x-axis**.

**Example:** Shift the following graph right by one unit.

y=2x

So take an input, say x=2, which gives us y=4, then if we are to shift the graph right, then the input of x=3 must give us the same output y=4

We must then reduce the input by one, so instead of inputting x into the equation, we must input x-1, which gives us a new equation of

y=2(x-1)

To translate left, the same thing must happen, except the number of units you’re shifting left will be added to the input.

**Remember **you add to the input to translate left, and you subtract from the input to translate right. It can be a little bit counter-intuitive.

**Vertical Translation**

**Vertical translations** move the graph up and down. This is along the** y-axis**.

Let’s shift a graph up by one unit, and let’s choose this equation:

y=2x

Take the input x=2 which has output y=4, to shift the graph up by 1 all we have to do is add another 1 to the output, so that when x=2 the output is y=5

This gives us the new equation:

y=2x+1,

and if we want to translate down we simply subtract by however much we want to translate down by.

**Translation by a Vector**

We may write a **translation in the form of a vector**

**\begin{pmatrix} a \\ b \end{pmatrix}**

where the graph is translated right by a units and up by b units. For example, if we are to translate the equation

y=x^2

by the vector we can begin by the right translation, so we must subtract the input by a

y=(x-a)^2

then the up translation, adding b to the output, giving us a new graph equation

y=(x-a)^2+b

**Reflections**

We will look at two** reflections**, a reflection over the **y-axis**, and a reflection over the **x-axis**. An easy way to visualise these is to imagine folding the graph along the axis and then imprinting the curve on the other side.

When reflected over the x-axis all positive y values become negative and all negative y values become positive, so the **output** of the equation must be **multiplied by -1**

For example y=\sqrt{x} reflected over the x-axis is y=-\sqrt{x}.

When reflecting over the y-axis all positive input (x) values become negative and all negative become positive. So the **input** must be multiplied by -1

For example y=3^x reflected over the y-axis is y=3^{-x}

The example picture is for y=(x-2)^2

**Example 1: Transformation of a Point**

Point A lies on a graph at coordinates (5, 6), find the coordinates of A after the following transformations

**a)** Translation by vector \begin{pmatrix} -3 \\ 2 \end{pmatrix}

**b)** Reflection over the y-axis

**c)** Reflection over the x-axis

**[6 marks]**

**a)** Firstly we recall that a vector translation tells us how it will move along the x-axis on the top row and how it will move along the y-axis on the bottom row. So in this case it translates the graph right by -3, or rather left by 3, and up by 2, which gives us

(5-3,6+2)=(2,8)

So the point corresponding to A is (2,8)

**b)** For a reflection over the y-axis the x variable is the one that becomes multiplied by -1, as the y-axis is vertical. So the new point, corresponding to A is (-5,6)

**c)** For a reflection over the x-axis the y variable is the one that becomes multiplied by -1, as the x-axis is horizontal. So the new point, corresponding to A is (5,-6)

**Example 2: Transformations of Equations**

The equation

y=\dfrac{1}{x}

is graphed. Find the equation for y after the following transformations

**a)** Translation by vector \begin{pmatrix} 4 \\ -2 \end{pmatrix}

**b)** Reflection over the y-axis

**[4 marks]**

**a)** With the vector we know that the graph is shifted right by 4 and down by 2, so to shift the equation right by 4 we must subtract the input by 4, giving us

y=\dfrac{1}{x-4}

Now, to shift down by 2 the output must be subtracted by 2, giving us a final equation of

y=\dfrac{1}{x-4}-2

**b)** For a reflection over the y-axis the input must be multiplied by -1, this gives us

y=\dfrac{1}{-x} = -\dfrac{1}{x}

Notice this is identical to a reflection on the x-axis.

## Graph Translations and Reflections Example Questions

**Question 1: **A graph has only one turning point, at (5, -3).

a) The graph is translated, the turning point is now (7, 2), what is the translation vector?

**[2 marks]**

b) The original graph is translated by vector

\begin{pmatrix} -2 \\ 3 \end{pmatrix}

What are the new coordinates of the turning point?

**[1 mark]**

a) If we say the translation vector is

\begin{pmatrix} a \\ b \end{pmatrix}

then this means that the new coordinate for the turning point is (5+a, -3 +b), so solving the x-coordinate

5+a=7

a=2

And solving the y-coordinate

-3 +b = 2

b=5

So the translation vector is

\begin{pmatrix} 2 \\ 5 \end{pmatrix}

b) The new coordinates will be

(5-2, -3 + 3)

(3,0)

**Question 2: **A graph shows the equation

y=x^2-2x

Find the equation for y, in its simplest form, after the following transformations

a) A translation by the vector\begin{pmatrix} 4 \\ 1 \end{pmatrix}

**[3 marks]**

b) A reflection over the y-axis

**[2 marks]**

a) The translation vector shifts the graph right by 4. This means 4 must be subtracted from the input. Which gives us

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

We also shift up by 1, so 1 must be added to the output

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

Which expands out to

y=x^2-10x+25

or

y=(x-5)^{2}

b) A reflection over the y-axis means the input must be multiplied by -1. Which gives us

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

As (-x)^2=x^2, we get

y=x^2+2x

**Question 3: **A graph shows the equation

y=3x^3-2x+3

Full describe the transformation from this equation to

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

**[2 marks]**

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

**[2 marks]**

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

**[2 marks]**

a) A reflection, over the y-axis.

b) A reflection, over the x-axis.

c) A translation by the vector \begin{pmatrix} 2 \\ -2 \end{pmatrix}