4.11.5 Horizontal Stretches and Compressions

Horizontal Transformations of Linear Functions

Horizontal transformations are the result of altering the input values before applying the function rule. Vertical transformations result from altering the output values after the function rule has been applied.

Example 1

Describe how $f(x)$ is transformed into $f(x-3)$.

Solution

First, graph the given function, $f(x)$, the parent linear function.

Now, let’s examine an input value of $x=2$ using $f(x)$.

• For the parent function $f(x)$, all points on the line are represented by $(x,f(x))$.
• So, when $x=2$, then $f(x)$ becomes $f(2)$.
• That means this point is located at $(2,f(2))$ or $(2,2)$.

In order to get the same output value for $f(x-3)$ as we had for $f(x)$, we need to figure out how to change the input value.

• Because our transformed equation is subtracting three from what we input, we have to increase the prior input value by 3 units to yield the same output as $f(x)$.
• The input value that is 3 units larger than $x=2$ is 5. In the transformed equation, we must use $x=5$.

So, in the original equation where $x=2$ resulted in $f(2)=2$, in the transformed equation, we have to use $x=5$ in order for $f(x-3)=f(5-3)=f(2)$.

On the graph, $x=5$ is located 3 units to the right of $x=2$. So, the transformed equation of $f(x-3)$ is 3 units to the right of $f(x)$.

Horizontal transformations are represented by the rule $f(x-c)$ where $c$ indicates the number of units to the right or left that the graph is shifted.

Example 2

Graph $f(x+2)$ when $f(x)=\frac{1}{2}x-3$. Then, determine the equation of the transformed function.

Solution

Step 1 - Graph the given function, $f(x)=\frac{1}{2}x-3$.

The y-intercept is (0, -3) and the slope is $\frac{1}{2}$.

For horizontal shifts, it is helpful to identify the x-intercept. On our graph, the x-intercept or zero is located at (6,0).

Step 2 - Apply the designated transformation. Remember that the rule for a horizontal shift is $f(x-c)$. If we rewrite $f(x+2)$ in an equivalent form, we can make it is easier to see $x-c$.

$f(x+2)=f(x-(-2))$ That means $c=-2$.

So, $f(x+2)$ represents a horizontal shift of two units to the left. On our graph, we move the x-intercept two units left from (6, 0) to (4, 0).

The orange dashed line is the graph of the original function and the blue solid line is the transformed function.

Step 3 - Determine the equation of the transformed function.

The y-intercept of the transformed line is (0, -2) and the slope of the line is $\frac{2}{4}$ or $\frac{1}{2}$ .

That means the equation $f(x+2)=\frac{1}{2}x-2$

Want to check your work? We can check our equation algebraically by substituting $x+2$ into $f(x)$.

$f(x)=\frac{1}{2}x-3$

$f(x+2)=\frac{1}{2}(x+2)-3=\frac{1}{2}x+1-3=\frac{1}{2}x-2$

Example 3

Graph $f(4x)$ when $f(x)=\frac{1}{2}x-3$. Then, determine the equation of the transformed function.

Solution

Step 3 - Graph the given function, $f(x)=\frac{1}{2}x-3$.

The y-intercept is (0, -3) and the slope is $\frac{1}{2}$ .

For horizontal shifts, it is helpful to identify the x-intercept. On our graph, the x-intercept or zero is located at (6,0).

Step 2 - Apply the designated transformation. Remember that the rule for a horizontal shift is $f(bx)$. And, when the value of $b$ is greater than 1, the function experiences a horizontal compression.

As a result of the horizontal compression, the linear function should look steeper because it is being compressed by a scale factor of 4.

On our graph, this means the x-intercept is compressed from (6, 0) to (1.5, 0).

The orange dashed line is the graph of the original function and the blue solid line is the transformed function.

Step 3 - Determine the equation of the transformed function.

The y-intercept of the transformed line is (0, -3) and the slope of the line is 3/1.5 or 2.

That means the equation $f(4x)=2x-3$.

Want to check your work? We can check our equation algebraically by substituting 4x into $f(x)$.

$f(x)=\frac{1}{2}x-3$

$f(4x)=\frac{1}{2}(4x)-3=2x-3$✔