Concept Byte: Collecting Quadratic Data

Use With Lesson 9-2

ACTIVITY

In this activity, you will use a loop of string to make rectangles, record their dimensions, and explore the graph of length versus area.

Activity

Take a piece of string, no more than 40 cm long, and tie the ends to form a loop. Use your thumbs and fingers to make a rectangle with the string. Hold it over a piece of graph paper to make right angles.

• Step 1 Copy the table below. Use the units on your graph paper to measure the length ℓ script l  and width w of the rectangle. Record the measurements to the nearest tenth of a centimeter.

  Length ( ℓ ) open script l close	Width (w)	Area (A)
  □ white square	□ white square	□ white square
  □ white square	□ white square	□ white square
  □ white square	□ white square	□ white square
  □ white square	□ white square	□ white square
  □ white square	□ white square	□ white square

• Step 2 Repeat Step 1 four more times, and complete the first two columns.
• Step 3 Record the area of each rectangle in the third column of the table.
• Step 4 Graph ordered pairs (length, area). Connect the points with a smooth curve.

Exercises

1. Writing Based on your graph from Step 4, explain why the data you collected in the activity cannot be modeled by a linear or exponential function.
2. 1. Find the length of your loop of string.
   2. Write an expression for the width of any rectangle made with your loop of string in terms of the rectangle's length ℓ . script l .
   3. Write a function for the area A of any rectangle made with your loop of string in terms of the rectangle's length ℓ . script l .
   4. Graph the function.
   5. Find the vertex of the graph. What is the meaning of the vertex?
3. Does your graph from part (d) of Exercise 2 exactly match the graph from Step 4 of the activity? If not, explain why not.
4. Suppose you repeated the activity with a loop of string that was 140 cm long. Write a function for the area A of any rectangle made with your loop of string in terms of the rectangle's length ℓ . script l .

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