5.4: Vertical Stretches & Compressions
Contents
Objectives
- Recognize that vertical stretches and compressions correspond with changes to the outputs
- Vertically stretch and compress a function that is given either explicitly or graphically
Important Items
Definitions:
vertical stretch, vertical compression
Lesson Guide
As you'll see below, there are two approaches for going about how to explain this section to your students. You should choose what will work best for your students.
Warm-Up
Depending on when this section occurs (in the past it has been after exam 2), it may be helpful to review all function transformation up to this point:
| Function | Transformation | Point | Input/output change? |
|---|---|---|---|
| f(x) | Base function | (-1,3) | N/A |
| f(x)+k | Shift f(x) up by k units | (-1,3+k) | Output |
| f(x)-k | Shift f(x) down by k units | (-1,3-k) | Output |
| f(x+h) | Shift f(x) left by h units | (-1-h,3) | Input |
| f(x-h) | Shift f(x) right by h units | (-1+h,3) | Input |
| -f(x) | Reflect f(x) across x-axis | (-1,-3) | Output |
| f(-x) | Reflect f(x) across y-axis | (1,3) | Input |
Have students do Problem 1. It may be worthwhile to discuss this as a review for everyone.
Recognize that vertical stretches and compressions correspond with changes to the outputs
Have students work on Problem 2. Discuss the differences students found in parts (b) and (c)
Observe that the $y$-intercept values change, but the $x$-intercepts stay the same, which makes sense since only the output is being changed.
Vertically stretch and compress a function that is given either explicitly or graphically
Work with students to fill this part out in their course packet (part (d)).
If $f(x)$ is a function and $k>1$ is a constant, then the graph of
*$g(x)=kf(x)$ [[vertically stretches} the graph of $f(x)$ by a factor of $k$,
* $g(x)=\frac{1}{k}f(x)$ [[vertically compresses} the graph of $f(x)$ by a factor of $k$.
*If $k<-1$, then the graph of $g(x)$ also involves a reflection of the graph of $f(x)$ about the $x$-axis.
Work through the rest of the problems on the worksheet, pausing to discuss as necessary. Make sure that everyone has the correct answer to Problem 5. There are a few different ways to think about it, but one thing to point out is that $x$-intercepts will never change due to a {\em vertical} stretch or compression. As a way to get them thinking ahead, ask what we might do to the graph that would change $x$-intercepts.
If time allows, let students work on the Synthesis Problem. This will likely be very difficult for students, but very cool if they see the trick! It's also similar to a problem on WeBWorK, so it is nice for them to have seen an example like this.
Comments
- I'm not entirely fond of how abruptly this lesson starts. I chose to motivate looking at compressions/stretches by talking about how we have added/subtracted constants to the outputs/inputs of a function, so what happens if we instead multiply/divide constants to the outputs/inputs of a function. -Elizabeth
NOTE: This comment was about the old lesson plan, but is also a good way to motivate the section!
