## One Graph, So Many Questions

I’m not sure where I first saw these types of questions. It was either at  Dan Greene’s1 or the Pre-AP Mathematics site2. Sadly, it was not in a textbook.

Here3 is one I made last week. I plan on using this with my freshmen sometime early first semester. There are a plethora of questions that can be asked, depending upon the concepts your students have learned. Some examples are:

• What is the midpoint of segment AB?
• What is the length of segment BC?
• What is the distance from A to C?
• What three dimensional figure is created when AB is rotated around the y-axis?
• Draw a segment from point B to point D. What is the area of triangle BDC?
• Draw the graph of f(x) + 2.
• What are the zeros of f(x)?
• What is the domain of f(x)?
• What is the range of f(x)?

So, you get the idea. What other questions can you come up with?

1 Dan Greene has some of the best stuff out there when it comes to multiple representations. I’ve used this on composite functions and this on transformations. Good stuff.
2 There are some nice activities at the Pre-AP site. I really like the idea of adapting released free-response questions too.
3 Here is the worksheet .doc or .pdf Only want the graph? Here you go .png

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### 12 Responses to One Graph, So Many Questions

1. H. says:

Great problem. As is your site makeover. Congrats!

2. Richard G says:

Jackie,

You could label the axes and see what happens. For instance:
X = time
Y = distance from Denver

And then change it to:

X = time
Y = speed in miles per hour

Just a thought.

3. Dan says:

Beautiful! I love it! 🙂

4. Jackie says:

Richard – I know what you’re getting at. We do tons of those problems. We decided we need to embed a few more “traditional” problems into IMP: state testing, college entrance exams, …

H & Dan – thanks!

5. Kate says:

If you do label the axes like Richard suggested, you could ask them to describe what is going on in each interval, and what is the significance of point C.

Write the equation of a line parallel to AB that passes through (3,1). Write the equation of a line perpendicular to AB that has a y-intercept of 4. Graph both of these.

What are the x-intercepts of f(x)?

What is the area under the curve? (Do you count the region below the x-axis as negative area? Interesting discussion.)

OK I’m tapped out. And planning on stealing this. 🙂 I did something kind of similar as an after-the-test project last year, but the emphasis was graphing lines and writing equations of lines instead of functions, and my questions weren’t nearly as varied.

6. Jackie says:

Nice additional questions Kate! There are so many that can be asked. The variation in questions is important. I think it’s a nice way to get in both review of concepts and preview of material they’ll cover in later courses. (area under the function – nice set up for integration, no?). Also the variation prevents the disconnect between different topics that occurs when we teach each concept in isolation (that “tell me what to do and I’ll do it” piece).

As for the stealing… that’s why it’s here. I plan to do more sharing this year. Really. One of my blog goals.

7. Brad Jolly says:

1) What is the range of the function?
2) What is the domain of the function?
3) Does the function have an inverse function? If so, draw it; if not, explain why it does not.
4) What is f(f(1))?
5) Suppose this is a graph of displacement vs. time. Draw the velocity vs. time graph.
6) Suppose this is a graph of velocity vs. time. Draw the displacement vs. time graph.
7) For what values of x is f(f(x)) undefined?
8) What is the area between the graph and the x-axis?
9) For what horizontal line y=k, where k is in the range of the function, is the area between the graph and the line y=k the greatest?
10) What is the total length of the line?
11) At what points does f'(x) = 0?
12) What is the arithmetic mean value of f(x) over the domain?
13) What are the equations of the four line segments?
14) Where is the function differentiable?
15) Where is the function continuous?

8. Brad Jolly says:

16) Suppose f(x) is an odd function. What is f(f(5))?
17) Suppose f(x) is an even function. What is f(f(5))?
18 ) Where does f(x) = x?
19) Other than the point found in problem 18, where does f(f(x)) = x?

9. Jackie says:

10. jd2718 says:

Things to sketch

f(x) + 2
f(x) – 2
f(x + 2)
f(x – 2)

aha!

2(f(x))
f(2x)

oh

f(-x)
-f(x)

hmm

Composition?

f(f(1))
f(f(2))

etc

and my favorites

f(|x|) (sort of boring here)
and
|f(x)|

I haven’t done this with composite functions, but I do it often with f(x) = a|x-h| + k

I like the idea of having kids make their own, show us the graph, and try to make us guess what f(x) is.

(Think of the letter W: (0,6), (2,0), (3,3), (4,0), (6,6) )

Jonathan

11. Dave Marain says:

Great stuff, Jackie…
You may want to examine current SAT-type questions that relate to the Algebra 2 strand (as well as the Algebra 2 End of Course Test from ADP). Here’s a fairly typical one:

From the graph above, for how many values of x is f(x) = 1?
(A) 0 (B) 1 (C) 2 (D) 3 (E) more than 3

There are endless variations on these. The f(f(…)) suggested by Brad is particularly nice and has been tested on recent SATs.
Dave

12. Jackie says:

Jonathan, great idea on the absolute value of f(x)! I’ll throw that in and some more of the transformations second semester (the questions I made are for a freshman class, we’ll do some previewing of Alg II topics 2nd semester).

Dave, the vast majority of my students will take the ACT not the SAT but those are still great questions. Thanks!