## Egg Launch Contest

- Lesson

Students will represent quadratic functions as a table, with a graph, and with an equation. They will compare data and move between representations.

In this activity, students encounter data that comes in different forms in the context of the description of an egg launch contest. The data for team A are shown in a table, the data for team B are expressed by an equation, and the data for team C are displayed in a graph. The data are available to students on the activity sheet.

- Have students read the first two paragraphs on the activity sheet.
Ask the class what they notice about the height of the egg as the
distance from the starting line increases. If the data points are
plotted on a coordinate plane and connected, what shape do students
think the graph makes?
[Students should notice that the height increases, then decreases. The shape is a parabola.]

- Have students read the third paragraph. Ask the class to describe the shape described by the equation.
[Students should recognize that this is a quadratic equation, whose graph is a parabola. The negative coefficient before the x

^{2}term means that the parabola opens down and has a maximum value.] - Have students read the fourth paragraph. Ask them what they know about the flight path of Team C's egg by looking at the graph.
- After a discussion of the starting points, the heights, and the distances from the starting point for the three teams, ask students to spend a minute on recording which team they think won the contest and why.

Put students in groups or pairs to work through the second page of the activity sheet. They will need a calculator or some other tool for regression to find the equations for Team A and Team C.

If students are using a TI–83/84, you can post instructions for:

Be sure to circulate around the classroom to help the students use the calculator effectively.

### Team A

- Equation: –1.3
*x*^{2}+ 39.6*x*– 195.1 - For the graph, see below.

### Team B

*Note that values are rounded and students may choose different points for their table.**x*2

3

6

9

12

15

18

21

22

*y*–5.2

9.8

45.2

66.2

72.8

65

42.8

6.2

–9.2

- For the graph, see below.

### Team C

*Note that values are rounded and students may choose different points for their table.**x*11

12

15

18

19

21

24

27

*y*0

19

65.5

86.5

88

(max)82

53

0

- Equation: –1.4
*x*^{2}+ 53.2*x*– 417

### Graph of All Functions

- Egg Launch Activity Sheet
- Graphing Calculator

**Assessment Option**

Ask students to write a news story that interprets the graphs of the flight paths of some of the other eggs in the contest such as the following:

Additional Graphs for Student Scenarios (in color and black-and-white)

**Extensions**

- Ask students to make colored posters of their graphs.
- Ask students to explore this scenario: "You have been asked to find a quadratic function. When graphed on the coordinate plane, the maximum height attained by the egg on its flight path is equal to the distance the egg is hurled down the field. Write such a quadratic function as an equation.

**Questions for Students**

- Describe the usefulness of each representation (table, graph, equation) of the data.
- What information about these egg launches can you infer from the tables, graphs, and equations?
- What effect do the negative leading coefficients of the equations have on the graphs?
- Explain different strategies that can be used to determine the maximum height reached by an egg on its flight path. What can you say about a minimum height reached by an egg on its flight path?
- What can you say about any symmetry in these graphs?

**Teacher Reflection**

- How did students adjust to the 3 representations of the data?
- Do students have a data representation preference? If so, what is their preference and why?
- How well did students use technology (calculators) to do some of the work for them? Describe what students did with the calculators to help them interpret the data.
- How did students react to the idea that the launcher did not need to be on the goal line of the football field?

### Learning Objectives

In this lesson, students will:

- Move between representations of a function as a table, a graph and an equation.
- Determine the maximum value of a quadratic function.
- Compare quadratic functions.

### NCTM Standards and Expectations

- Analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior.

- Understand the meaning of equivalent forms of expressions, equations, inequalities, and relations.

- Draw reasonable conclusions about a situation being modeled.