When you look at the graph of h(x), how does the y-intercept differ from f(x)? What transformation could cause this shift?
Transformations: h(x) is a vertical stretch by a factor of 4, followed by a vertical translation 2 units down. Graph:
Practice makes perfect
Let's begin by graphing both functions. Then, we can compare the differences between them to determine what transformations may have taken place.
First, look at how the slope changes between f(x) and h(x).
Looking at the slope triangles, we can see that the rise of h(x) is 4 times greater than the rise in f(x). This means that h(x) is moving away from the x-axis 4 times as quickly as f(x).
Slope off(x):&1 ⇔ rise/run=3/3 [0.8em]
Slope ofh(x):&4 ⇔ rise/run=12/3
When a function is pulled away from or pushed towards the x-axis, this is either a vertical stretch or shrink. Because the transformed function is farther from the x-axis, it is a vertical stretch by a factor of 4. The next thing to notice is how the y-intercept changes.
The y-intercept moved 2 units down. Any shift in the vertical or horizontal direction is a translation. This is a vertical translation down by 2.
