Constructing Parabolas

These web-based activities were developed, during summer sessions from 1997 through 1999, by students at Mainland High School, located in Daytona Beach, FL. As part of Mainland's CCTT grant, the students were given permission by the book's publisher to convert the ancillary materials of the County's adopted Algebra 1 textbook (Glencoe Algebra 1 — ISBN: 9780028253282) into html format.

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.