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 $h (x) = f (c x)$ 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 $y -$ 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 $c = - 2 .$ This is also a horizontal shrink but, additionally, it is a reflection in the $y -$ 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 $y -$ axis. Therefore, the function is growing slower than the original function.

Finally, we can see what happens when $c = - \frac{1}{2} .$ 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 $y -$ axis. Notice how it is a flipped version of the previous graph.

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