Another comment on the Linkedin blog “Math, Math Education, Math Culture”

I replied to David’s post:

When I am teaching graphs of quadratic functions, I like to ask my students why we don’t also stretch the graph horizontally. The same question can be asked for exponential and logarithmic functions. Unfortunately, this question usually goes over their heads. (This is probably my fault.)

Here is my reply:

Actually, the graph is stretched horizontally when the x is multiplied by a factor. If the factor is >1, the graph is shrunk horizontally by the reciprocal of the factor. If the factor is < 1, the graph is stretched horizontally by the reciprocal of the factor.

This phenomenon is most easily seen in a simple graph, such as y = abs(x).

The graph is in the form of a V going with the vertex at the origin. To the right of the origin, the graph is the line y = x. To the left of the origin is the line y = -x. The domain is all real numbers and the range is y >= 0.

I did this on a graphing calculator. I let Y1 = abs (x), Y2 = abs (2x), and Y3 = abs (.5x)

When discussing changes in the domain, I note the changes in the x-value from one graph to the other.

When discussing the changes in the range, I note the changes in the y-value from one graph to the other.

I drew the horizontal lines y = 1 and y = 2 to illustrate the connections between the graphs.

The intersection of line y = 1 and y = abs (x) is at (-1, 1) and (1,1)

y = 1 and y = abs(2x) is at (-.5, 1) and (.5, 1)

y = 1 and y = abs (.5x) is at (-2,1) and (2,1)

Notice that the range didn’t change (all of the y-values remained at 1).

However the domain did change.

The domain value of abs(2x) was half of the domain value of abs (x)

The domain value of abs (.5x) was twice the domain value of abs (x)

It seems like the multiplier has a reciprocal effect on the domain. Let’s look some more to verify that.

The intersection of line y = 2 and y = abs (x) is at (-2, 2) and (2,2)

y = 2 and y = abs(2x) is at (1, 2) and (1, 2)

y = 2 and y = abs (.5x) is at (-4,2) and (4,2)

The range value didn’t change (all of the y-values remained at 2)

The domain value of abs (2x) was half the domain value of abs (x)

The domain value of abs (.5x) was twice the domain value of abs (x)

We are seeing the same reciprocal affect. When the domain is multiplied by a number greater than 1, say “a”, the domain multiplied by the reciprocal of “a”, which is “1/a”, which is smaller than 1, hence shrinking the domain by a factor of 1/a

When the domain is multiplied by a number less than 1, let’s call it “b”, then the domain is multiplied by the reciprocal of “b” which is “1/b”. However, since b <1, “1/b” will be greater than 1. This will cause the domain to stretch by a factor of 1/b.

Although the stretch may look vertical, the reason it is changing is not due to the range (vertical) element changing while the domain element (horizontal) remains the same. The “stretch” is really horizontal, with the domain (horizontal) element changing while the range element (vertical) remains the same.

After posting the above, there was a post by Jered who said the following:

Elaine: The problem with this set of examples is that

• Y2 = |2x| = 2|x| and

• Y3 = |0.5x| = 0.5|x|.

In other words, we may equally view Y2 as a horizontal compression or as a vertical stretch, and Y3 as a horizontal stretch or a vertical compression!

This is why linear functions, absolute-value functions, and quadratic functions all are not such great illustrations of the difference between horizontal and vertical dilations.

Here is his reply:

Elaine: Well, of course trigonometric functions make really nice examples for this, but those often aren’t available at the time we’re trying to introduce this topic. (It happens that when I last taught the subject it *was* for the purpose of understanding transformations of sin/cos functions.)

Since your students seem to be using graphing calculators, you can use combinations of common functions: for example,

• Y1 = x² + abs(x)

• Y2 = (2x)² + abs(2x)

• Y3 = 2(x² + abs(x)).

This way Y2≠Y3 (i.e. the vertical stretch really is different from the horizontal compression), but at the cost of examining a somewhat more complicated function.

David Radcliffe • Yes. We need four parameters to specify a general sine wave:

y = A sin(B(x-C)) + D.

It is very interesting that for quadratics and some other classes of functions (such as absolute value, exponential, and logarithmic functions) only three parameters are required, because a horizontal compression is equivalent to a vertical stretch. This shows that these curves have a continuous family of non-rigid symmetries. Unfortunately, this is way too advanced for my students, or more likely, I lack the skill to explain it in simpler terms.

Jered Wasburn-Moses • Elaine: You’re welcome, but after actually looking I’m not so sure it’s a great example. I tried it on a TI-84, and it’s really very difficult to see what’s going on. It’s just not clear enough that there’s any real difference in the shapes between Y2 and Y3–they look like continuations of the same dilations.

I tried replacing either of the two functions with sqrt(), but with similar results.

Back to the drawing board!

Elaine Watson • Jered: I like your perseverance! I’m going to go work in my garden and clear my head! I’ll be thinking about examples as well…

Jered Wasburn-Moses • Well, Elaine, I must admit–this topic is *far* more interesting than trying to figure out my summer staffing schedule, which is what I’m ostensibly working on!

Jered Wasburn-Moses • Okay, I don’t know how you’ll feel about this, but what about:

• Y1 = x³ – 2x² – x + 2

• Y2 = (2x)³ – 2(2x)² – (2x) + 2

• Y3 = 2(x³ – 2x² – x + 2)

The advantage is that the transformations are pretty visible. The downside is that Y2 in particular may be difficult for students to follow.

// Addendum: I recommend a viewing window of about x: [-2, 3], y: [-5,5].

Elaine Watson • Jered,

I love it! It’s very clear looking at the zeros of Y1 and Y2 that the domain is affected and also very clear that the range is not affected.

Same with Y1 and Y3, it’s clear when you look at the max’s and min’s that the range is affected, but the domain stays the same.

Thanks for your perseverance and I’m glad that I could provide you with a distraction to figuring out your summer staffing schedule. What is your role? Are you an administrator?

One reason that I’m posting this discussion from another blog is that I am fascinated by the “crowd sourcing” available. I put out something in response to a comment. My comment was improved upon. In the end, everyone benefited from one member of the discussion taking it further.

Math teachers and professionals don’t often have time to meet face to face and discuss math issues, compare notes, or critique each other’s reasoning. The blog platform is a wonderful way to expand our learning communities to the global community.