The Math Forum || Annie's Sketchpad Activites || Printable Version (No Java)

Three Roads to a Parabola

In this lesson we will explore a conic section called the parabola. This shape shows up a lot in the world - it is the path of a ball that is thrown into the air, and it is the shape of automobile headlights and searchlights.

You'll need a piece of paper suitable for folding (clean unlined paper or patty paper).

Road 1: Paperfolding

Take your piece of paper and put a dot on it near one edge. Now fold that closest edge so that it touches the point. Do this about 20 times, unfolding the paper after each fold and then using a new point on the edge.

You should begin to see a curve appearing on your paper. The lines you have created are tangent to the curve. Compare your curve to that of the person next to you. In what ways are they the same? In what ways are they different?

What happens when the point moves closer to the bottom edge? What happens when it moves further away?

 

Road 2: Sketchpad Construction

Now let's model our paper folding construction with Sketchpad. In a new sketch, draw a long segment near the bottom of your window. This is the bottom of your paper. Place a point somewhere above the segment (C). This is the point on the paper. Select the segment and choose Point on Object from the Construct menu (D). We need to construct the line that represents the fold when C and D match up.
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Look back at your paper construction. Fold along one of the lines and make a mark on the bottom edge where it hits the point. This gives you two points to work with. How is the fold related to those two points? (You might want to try this on a fresh piece of paper to get a clearer view.) Construct this line in Sketchpad.

Now that you have your line, make sure it's selected and go to the Display menu and choose Trace Line. If you drag D along AB, you'll get your curve!

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Move C closer to AB. Drag D again.

We could continue doing this, but it would be really slick to have D animated along AB so that we didn't have to drag it every time. To do this, select D and the segment AB (not the endpoints, just the segment). From the Edit menu, choose Action Button->Animation. Click Animate. There is now an Animate button on the screen. Double click this. Pretty cool, eh? (You can click anywhere on the screen to get the animation to stop.)

What happens when C is closer to AB? What happens when it's far away? Is this the same thing you found with the paper folding?

 

 

It would be a lot easier to see the differences if our traced curve didn't disappear every time we clicked on the screen to move something. Instead of tracing the line, we can construct the locus. Select the line and point D and choose Locus from the Construct menu. Now you can move C or AB and the envelope will stay in place.

What we would really like, though, is the actual points on the curve. The lines we're tracing are the tangents to the curve, and the points we want lie on those lines somewhere. Let's see if we can find them.

The definition of a parabola states that a curve is a parabola if, given any point on the curve, the distance from that point to the focus is the same as the distance from that point to the directrix. Our segment is our directrix here, and the point C is the focus. How can we find the points that fill this criteria? We know they're on the lines somewhere. Here's a hint: think about how you find the distance from any point to a line. Here's another hint: it only takes one more line in your sketch. (You might get together with your neighbor to figure this part out.)

How did you find the point? Explain why you think this point is equidistant from the focus and the directrix.

 

We could trace this point to see the curve, but it's still going to go away when we click since the path is only traced, not constructed. It would be awesome to have this curve as an object that moves dynamically when we drag D.

To do this, hide the objects in the sketch that you don't need (keep the focus, directrix, D, and your new point). Select the new point and D and choose Locus from the Construct menu - we want the locus of the new point when D moves. Your screen should look something like this:

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Now move C. What happens to the curve? Move AB. What happens if you move AB above C?

 

Road 3: Graphing the Equation of a Parabola

The previous two sections looked at a parabola from its definition: the set of points equidistant from a focus and a directrix. In this section we'll graph one analytically.

Open a new sketch and choose Create Axes from the Graph menu. We're going to graph the equation f(x)=ax^2 + b. To do this, we need three variables, a, b, and x. We're going to use the x-coordinates of three points on the x-axis to serve as the variables.

Put three points on the x-axis. Make sure they aren't attached to the unit point or the origin - they need to be free (aside from being attached to the x-axis). Label these points A, B, and X. Select the three points and choose Coordinates from the Measure menu. We need the x-coordinates of these points. Double-click on the coordinates of A. In the calculator, hold down the Values menu, slide down to Point A, and slide over to x. Click okay. You've now got the x-coordinate of point A. Move A along the axis to make sure this number changes. Do the same for B and X. Hide the coordinates of the three points (don't delete them!).

Now we need the equation for f(x), ax^2 + b. Select the x-coordinates for A, B, and X and open the calculator. Under the values menu, choose x[A]. Click the * for multiplication. Choose x[X] from the values menu. Click ^ and 2. Click +. Choose x[B] and click okay. You should have something like this:

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Again, move A, B, and X to ensure that everything changes.

Choose xX and your equation and go to the Graph menu and choose Plot as (x, y). You should get a point on your screen. If you can't see your point, move A, B, and X closer to the origin - your value for f(x) may be too large to see on your axes.

Let's construct the locus of your new point as X moves along the x-axis. Choose the new point and X. Choose Locus from the Construct menu. Awesome!

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What happens when you move B?

What happens when you move A? What if A is negative? What if it's zero?

What happens if, instead of picking xX and the equation, you switch the order - choose the equation and xX. Now plot the point and construct the locus. What happens to the shape?

We could have gotten three variables in a number of ways. The first time I did this construction I used the lengths of line segments. What's the disadvantage of doing this?

Try repeating the construct with the equation f(x)=a(x-h)^2 + k. How do a, h, and k affect the shape and location of the parabola?