Exercise 4 Page 145

We are given our base function for comparison, $f(x)=x.$

Let's consider the graphs of $g(x)$ and $h(x)$ compared to $f(x)$ one at a time.

Function g(x)

The function $g(x)=f(x)+c$ tells us that we are taking each output of the function $f(x)$ and adding $c.$

Notice that if $c$ is a negative number, it will be more like we are subtracting $c.$

These types of transformations are considered vertical translations.

Function h(x)

The function $h(x)=f(cx)$ tells us that we are taking each input of the function $f(x)$ and multiplying it by $c.$ There are several things to consider, such as whether $c$ is positive or negative and whether $|c|>1$ or if $0<|c|<1.$ First, let's look at the case where $c=2.$ This is going to be considered a horizontal shrink by $\frac{1}{2}.$

The above graph is considered a horizontal shrink because $h(x)$ is shrunken closer to the $y\text{-}$axis. This means that the function is growing quicker at lower values of $x.$ Next, let's look at what happens in the case where $c=\text{-}2.$ This is also a horizontal shrink but, additionally, it is a reflection in the $y\text{-}$axis. Notice how it is a flipped version of the previous graph.

Now, let's look at the case where $c=\frac{1}{2}.$ This is going to be considered a horizontal stretch by a factor of $2.$ It is considered a stretch because $h(x)$ is stretching farther away from the $y\text{-}$axis. Therefore, the function is growing slower than the original function.

Finally, we can see what happens when $c=\text{-}\frac{1}{2}.$ This is also going to be a horizontal stretch but, like the last time we had a negative value for $c,$ it is also a reflection in the $y\text{-}$axis. Notice how it is a flipped version of the previous graph.