Vertical stretches and reflections

So far, we have seen the possible effects of adding a constant value to function output and adding a constant value to function input . Each of these actions results in a translation of the function’s graph (either vertically or horizontally), but otherwise leaving the graph the same. Next, we investigate the effects of multiplication the function’s output by a constant.

We summarize and generalize our observations from the graphs above as follows.

Consider the functions and given in the following graphs.

a.
On the same axes as the plot of , sketch the following graphs: and . Be sure to label several points on each of , , and with arrows to indicate their correspondence. In addition, write one sentence to explain the overall transformations that have resulted in and from .
b.
On the same axes as the plot of , sketch the following graphs: and . Be sure to label several points on each of , , and with arrows to indicate their correspondence. In addition, write one sentence to explain the overall transformations that have resulted in and from .
c.
On the additional copies of the two figures below, sketch the graphs of the following transformed functions: (at left) and . As above, be sure to label several points on each graph and indicate their correspondence to points on the original parent function.
d.
Describe in words how the function is the result of three elementary transformations of . Does the order in which these transformations occur matter? Why or why not?

Horizontal Stretches

Follow the Link to Desmos https://www.desmos.com/calculator/xjem27frqi.
(a)
Make sure that the following graphs are enabled.

What effect do the 1.5, 2, 0.5 and 0.25 seem to have?

(b)
Now disable the previous graphs and make sure that the following graphs are enabled.

What effect do the 1.5, 2, 0.5 and 0.25 seem to have?

Reflections Across Axes

Points and are reflections of each other across the x-axis. Points and are reflections of each other across the y-axis. In general, two points that are symmetric with respect to a line are reflections of each other across that line.

Notice that this is just a special case of horizontal or vertical stretching where the factor we are multiplying by is !