guide to function transformations

For a given function $f(x)$, transformations can change the graph.

This guide will focus on the simple transformations of a function - dilation, reflection and translation.

Firstly, think about a point, $(a,b)$ on the plane. After it undergoes all the transformations it wants, we end up with a point, $(a^{\prime },b^{\prime })$.

So essentially, two variables, we do things, and end up with two variables. This is a function with two inputs and two outputs - $t(a,b)=\{g(a,b),h(a,b)\}$

• Worked example: The point $(2,3)$ is transformed using the function $t(a,b)=\{a+1,-2b\}$, Resulting in $(a^{\prime },b^{\prime })$. Find $(a^{\prime },b^{\prime })$.

Directly using the function, we see that $t(2,3)=\{3,-6\}$ and $(a^{\prime },b^{\prime })=(3,-6)$

• Worked example: The function $y=x^2$ is dilated by factor of 3 from the x-axis. Find the resulting image function, $y^{\prime }$.

A dilation of 3 from the x-axis means $y^{\prime }=3y$.

In the beginning we have $y=x^2$
In the end we have $y^{\prime }=x^{\prime }$

$x^{\prime }=x$
$y^{\prime }=3y$

So $y^{\prime }=3x^2$

• Worked example: The function $y=3x^2$ is translated to the right by 7 units. Find the resulting image function, $y^{\prime }$.

A translation to the right means $x^{\prime }=x+7$

In the beginning we have $y=3x^2$
In the end we have $y^{\prime }=x^{\prime }$

$x^{\prime }=x+7$
$y^{\prime }=y$

So $y^{\prime }=3x^2+7$

• Worked example: The function $y=x^2$ is translated upwards by 7 units. Then it is reflected across the x-axis. Find the resulting image function, $y^{\prime }$.

A translation upwards means $y^{\prime }=y+7$
A reflection in the x-axis means $y^{\prime }=-y$

In the beginning we have $y=x^2$
In the end we have $y^{\prime }=x^{\prime }$

original
$y^{\prime }=y$
translate
$y^{\prime }=y+7$
reflect
$y^{\prime }=-y-7$

So $y^{\prime }=-x^2-7$

• Worked example: The function $y=x^2$ is translated to the left by 6 units. Then it is dilated by factor of $\frac{1}{3}$ in the y-axis. Find the resulting image function, $y^{\prime }$.

A translation to the left means $x^{\prime }=x-6$
A dilation in the y-axis means $x^{\prime }=\frac{1}{3}x$

In the beginning we have $y=x^2$
In the end we have $y^{\prime }=x^{\prime }$

original
$x^{\prime }=x$
translation
$x^{\prime }=x-6$
dilation
$x^{\prime }=\frac{1}{3}x-2$

So $y^{\prime }=\frac{1}{3}x-2$

• Worked example: The function $y=x^2$ is translated

• Worked example: State the sequence of transformations needed to transform $y=x^2$ into the image function $y^{\prime }=2x^2+4x+2$

break down terms of $x$ - $2x^2+4x+2=2(x^2+2x+1)=2(x+1{)}^2$

original
$y=x^2$
dilation
$y=2x^2$