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Algebra 1

7.6.3 Using Quadratic Functions to Describe Height

Algebra 17.6.3 Using Quadratic Functions to Describe Height

7.6.3 Using Quadratic Functions to Describe Height

7.6.3 • Using Quadratic Functions to Describe Height

Activity

1. The function defined by d = 50 + 312 t − 16 t 2 d = 50 + 312 t − 16 t 2 gives the height in feet of a cannonball t t seconds after the ball leaves the cannon.

a. What does the term 50 tell us about the cannonball?

b. What does the term 312 t 312 t tell us about the cannonball?

c. What does − 16 t 2 − 16 t 2 tell us about the cannonball?

2. Use the graphing tool or technology outside the course. Graph the function. Adjust the graphing window to the following boundaries: 0 < x < 25 0 < x < 25 and 0 < y < 2000 0 < y < 2000 .

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3. Observe the graph and:

a. Describe the shape of the graph. What does it tell us about the movement of the cannonball?

The shape of the quadratic function is called a parabola. It has a “U” shape.

b. Estimate the maximum height the ball reaches. When does this happen?

The peak of the parabola, where the maximum occurs, is also called the vertex of the graph.

c. Estimate when the ball hits the ground.

The zero of a function is where the function equals 0, or where it crosses the x x -axis. In this case, there is only one zero where the cannonball hits the ground. The parabola does cross the x x -axis in the negative x x -values, but that does not make sense in this problem.

4. What domain is appropriate for this function? Be prepared to show your reasoning.

Are you ready for more?

Extending Your Thinking
1.

If the cannonball were fired at 800 feet per second, would it reach a mile in height? Remember there are 5280 feet in a mile. Be prepared to show your reasoning.

Video: Using Quadratic Functions to Describe Height

Watch the following video to learn more about quadratic functions.

7.6.3

Self Check

A toy rocket is launched from a platform 4 feet above the ground at a velocity of 128 feet per second. Its height is graphed below. Approximately what is the rocket’s maximum height?

  1. 128
  2. 110
  3. 260
  4. 8

Additional Resources

Reading Graphs of Quadratics

An object is thrown upward from a height of 5 feet with a velocity of 60 feet per second. Its height, h ( t ) h ( t ) , in feet after t t seconds is modeled by the function h ( t ) = 5 + 60 t − 16 t 2 h ( t ) = 5 + 60 t − 16 t 2 .

Step 1 -Identify the starting height.

5 feet

Step 2 -Identify the maximum height by looking at the vertex or peak of the parabola.

approximately 62 feet

Step 3 -Identify when the object reaches its maximum by matching the maximum with its matching time from the x x -axis.

approximately 1.75 seconds

Step 4 -When does the object hit the ground?

Look at where the object has a height of 0 feet, or the zero of the function. This happens at approximately 3.75 seconds. Notice there would be another zero on the graph of the function to the left of x = 0 x = 0 , but that does not make sense in the problem because the object was thrown from 5 feet.

Try it

Reading Graphs of Quadratics

A rock is launched into the air, and its height is represented by the graph below.

What is the maximum height of the rock?

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