Investigating a Quadratic Relationship

Introduction

We will be looking at how mathematics can model a real life situation. In this activity, we will plot some data on to the coordinate plane and then estimate some of the values that are not in the data set. We will then work to fit an equation to the data and then check our estimates against the equation.

A diver jumps off a springboard from a height of 3 meters off the water.

The Data

The table below shows the diver's height in meters as a function of time in seconds:

Plot the ordered pairs that correspond to diver's height and time. (Use the Point tool or input the ordered pairs)

What is her rate of change (speed) during the first 0.2 seconds of her jump? Use the Line and Slope tools or write an equation. Then write your solution below.

How fast is she going during the NEXT 0.2 seconds? Use the Line and Slope tools or write an equation. Then write your solution below.

About what time does she reach the top of her jump? Explain your reasoning. Hint: What will be her speed at that time?

Now the points are plotted for you. Use the line tool to connect the points.

Now use the graph to estimate what time she hits the surface of the water. Hint: What will be her height at that time?

Use your estimate to determine how fast she is moving when she hits the water. Use the Line and Slope tools or write an equation. Then write your solution below.

Move the sliders for a, b, and c to find a quadratic equation that matches the graphed points.

Since the diver's height (h) is a function of time (t), we can model her motion with the basic quadratic equation: h(t)=at²+bt+c Based on the values you found for a, b, and c in your graph, what is the equation that fits her motion? Write your equation below. Remember, you can right click on the orange line to display your equation.

Now use your equation and find: a) the time she reaches the top of her jump b) the time she hits the water c) her speed as she hits the water

Describe how your estimated answers above compare with the results the equation gave you.