6. Transformations of Graphs of Linear Functions
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See solution.
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.
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.
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 12.
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= 12. 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=- 12. 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.
