Constructing Parabolas | CCTT MathSummer

Constructing Parabolas

Through our lessons you have seen that a parabola is the graph of a quadratic function. In this activity you will construct a parabola without the use of an equation.

Materials

heavy construction paper

string

tape

scissors

pencil

ruler or straightedge

looseleaf paper

Copy the L-square shown below onto heavy construction paper and cut it out.

Next tape a piece of string at point A. Using the string, measure the distance AY. Cut the string at Y.

Now draw a straight line on a sheet of looseleaf paper and place your L-square along the straight line.

Choose any point P and tape the free end of the string at point P.

Next place your pencil at point D along the L-square and use your pencil to hold the string tightly against the L-square.

1. What happens as you move the L-square to the left along the line?

2. Now choose point P at a different location. Repeat the process. Describe what happens to the curve as the location of point P changes.

You can also draw a parabola on "conic" graph paper, which is made by drawing a set of concentric circles on lined paper. Use this grid for exercises 3 and 4.

Mark the point where line 1 intersects circle 1. Next mark the two points where line 2 intersects circle 2. Then mark the points where line 3 intersects circle 3. Repeat the process for lines and circles 4 and 5. Connect the points with a smooth curve.

3. Describe your graph.

On the same conic grid, mark the point where line 0 intersects circle 2. Next mark the two points where line 1 intersects circle 3. Then mark the two points where line 2 intersects circle 4. Repeat the process for lines 3 and 4. Connect the points with a smooth curve.

4. Describe your graph.

A nomograph is a chart used to turn difficult calculations into simple ones. The graph of a quadratic function can be used to make a nomograph for finding one side of a right triangle when 2 are given. This nomograph is based on the "Pythagorean" Right-Triangle Theorem. It makes extensive use of squares and square roots.

To find the hypotenuse of a right triange in which side a is 30 units and side b is 40 units, locate 30 on the a scale and lay a straightedge from 30 on the a scale to 40 on the b scale. The stright line crosses the (c) scale at 50. Thus the hyptenuse c of the right triangle with sides 30 and 40 is 50.

5. Use the nomograph to complete the table.

a b c

3 4 c

a 35 37

8 b 17

6. The values in the table above are called Pythagorean Triplets. Describe the relationship you notice about a² + b² and c².