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Created page with " Prior Lesson | Next Lesson ==Objectives== * Recognize that vertical stretches a..."

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[[5.3: Reflections & Even and Odd Functions | Prior Lesson]] | [[5.5: Horizontal Stretches & Compressions | Next Lesson]]
==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:

{| class="wikitable"
|+ Caption: example table
|-
! 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!

5.4: Vertical Stretches & Compressions - Revision history

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives== * Recognize that vertical stretches a..."