# 3.5 Transformation of functions  (Page 3/21)

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Given a tabular function, create a new row to represent a horizontal shift.

1. Identify the input row or column.
2. Determine the magnitude of the shift.
3. Add the shift to the value in each input cell.

## Shifting a tabular function horizontally

A function $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ is given in [link] . Create a table for the function $\text{\hspace{0.17em}}g\left(x\right)=f\left(x-3\right).$

 $x$ 2 4 6 8 $f\left(x\right)$ 1 3 7 11

The formula $\text{\hspace{0.17em}}g\left(x\right)=f\left(x-3\right)\text{\hspace{0.17em}}$ tells us that the output values of $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ are the same as the output value of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ when the input value is 3 less than the original value. For example, we know that $\text{\hspace{0.17em}}f\left(2\right)=1.\text{\hspace{0.17em}}$ To get the same output from the function $\text{\hspace{0.17em}}g,\text{\hspace{0.17em}}$ we will need an input value that is 3 larger . We input a value that is 3 larger for $\text{\hspace{0.17em}}g\left(x\right)\text{\hspace{0.17em}}$ because the function takes 3 away before evaluating the function $\text{\hspace{0.17em}}f.$

$\begin{array}{ccc}\hfill g\left(5\right)& =& f\left(5-3\right)\hfill \\ & =& f\left(2\right)\hfill \\ & =& 1\hfill \end{array}$

We continue with the other values to create [link] .

 $x$ 5 7 9 11 $x-3$ 2 4 6 8 $f\left(x–3\right)$ 1 3 7 11 $g\left(x\right)$ 1 3 7 11

The result is that the function $\text{\hspace{0.17em}}g\left(x\right)\text{\hspace{0.17em}}$ has been shifted to the right by 3. Notice the output values for $\text{\hspace{0.17em}}g\left(x\right)\text{\hspace{0.17em}}$ remain the same as the output values for $\text{\hspace{0.17em}}f\left(x\right),\text{\hspace{0.17em}}$ but the corresponding input values, $\text{\hspace{0.17em}}x,\text{\hspace{0.17em}}$ have shifted to the right by 3. Specifically, 2 shifted to 5, 4 shifted to 7, 6 shifted to 9, and 8 shifted to 11.

## Identifying a horizontal shift of a toolkit function

[link] represents a transformation of the toolkit function $\text{\hspace{0.17em}}f\left(x\right)={x}^{2}.\text{\hspace{0.17em}}$ Relate this new function $\text{\hspace{0.17em}}g\left(x\right)\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}f\left(x\right),\text{\hspace{0.17em}}$ and then find a formula for $\text{\hspace{0.17em}}g\left(x\right).$

Notice that the graph is identical in shape to the $\text{\hspace{0.17em}}f\left(x\right)={x}^{2}\text{\hspace{0.17em}}$ function, but the x- values are shifted to the right 2 units. The vertex used to be at (0,0), but now the vertex is at (2,0). The graph is the basic quadratic function shifted 2 units to the right, so

$g\left(x\right)=f\left(x-2\right)$

Notice how we must input the value $\text{\hspace{0.17em}}x=2\text{\hspace{0.17em}}$ to get the output value $\text{\hspace{0.17em}}y=0;\text{\hspace{0.17em}}$ the x -values must be 2 units larger because of the shift to the right by 2 units. We can then use the definition of the $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ function to write a formula for $\text{\hspace{0.17em}}g\left(x\right)\text{\hspace{0.17em}}$ by evaluating $\text{\hspace{0.17em}}f\left(x-2\right).$

$\begin{array}{ccc}\hfill f\left(x\right)& =& {x}^{2}\hfill \\ \hfill g\left(x\right)& =& f\left(x-2\right)\hfill \\ \hfill g\left(x\right)& =& f\left(x-2\right)={\left(x-2\right)}^{2}\hfill \end{array}$

## Interpreting horizontal versus vertical shifts

The function $\text{\hspace{0.17em}}G\left(m\right)\text{\hspace{0.17em}}$ gives the number of gallons of gas required to drive $\text{\hspace{0.17em}}m\text{\hspace{0.17em}}$ miles. Interpret $\text{\hspace{0.17em}}G\left(m\right)+10\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}G\left(m+10\right).$

$G\left(m\right)+10\text{\hspace{0.17em}}$ can be interpreted as adding 10 to the output, gallons. This is the gas required to drive $\text{\hspace{0.17em}}m\text{\hspace{0.17em}}$ miles, plus another 10 gallons of gas. The graph would indicate a vertical shift.

$G\left(m+10\right)\text{\hspace{0.17em}}$ can be interpreted as adding 10 to the input, miles. So this is the number of gallons of gas required to drive 10 miles more than $\text{\hspace{0.17em}}m\text{\hspace{0.17em}}$ miles. The graph would indicate a horizontal shift.

Given the function $\text{\hspace{0.17em}}f\left(x\right)=\sqrt{x},\text{\hspace{0.17em}}$ graph the original function $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ and the transformation $\text{\hspace{0.17em}}g\left(x\right)=f\left(x+2\right)\text{\hspace{0.17em}}$ on the same axes. Is this a horizontal or a vertical shift? Which way is the graph shifted and by how many units?

The graphs of $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)\text{\hspace{0.17em}}$ are shown below. The transformation is a horizontal shift. The function is shifted to the left by 2 units.

## Combining vertical and horizontal shifts

Now that we have two transformations, we can combine them. Vertical shifts are outside changes that affect the output ( y -) values and shift the function up or down. Horizontal shifts are inside changes that affect the input ( x -) values and shift the function left or right. Combining the two types of shifts will cause the graph of a function to shift up or down and left or right.

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