Unit 7, Lesson 5
Building Quadratic Functions to Describe Situations (Part 1)
Algebra 1
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A rock is dropped from the top floor of a 500-foot tall building. A camera captures the distance the rock traveled, in feet, after each second.
An image of a rock dropped from the top floor of a 500-foot tall building. A camera displays the distance the rock traveled, in feet, after each second. The displays are 0 seconds comma fell 0 feet, 1 second comma fell 16 feet, 2 seconds comma fell 64 feet, 3 seconds comma fell 144 feet, 4 seconds comma fell 256 feet and 5 seconds comma fell 400 feet.
She wrote down:
Then, she noticed that 1, 4, 9, 16, and 25 are and .
The distance traveled by a falling object in a given amount of time is an example of a quadratic function. Galileo is said to have dropped balls of different mass from the Leaning Tower of Pisa, which is about 190 feet tall, to show that they travel the same distance in the same time. In fact the equation models the distance , in feet, that a metal ball falls after seconds, no matter what its mass.
Because , and the tower is only 190 feet tall, a metal ball hits the ground before 4 seconds.
Here is a table showing how far a metal ball has fallen over the first few seconds.
| time (seconds) | distance fallen (feet) |
|---|---|
| 0 | 0 |
| 1 | 16 |
| 2 | 64 |
| 3 | 144 |
Here are the time and distance pairs plotted on a coordinate plane:
Graph of the quadratic function
on a coordinate plane, origin
. Horizontal axis scale 0 to 6 by 1’s, labeled “time (seconds)”. Vertical axis scale 0 to 200 by 50’s, labeled “distance dropped (feet)”. The four points of the function shown are
(0 comma 0), (1 comma 16), (2 comma 64) and (3 comma 144).
Notice that the distance fallen is increasing each second. The average rate of change is increasing each second, which means that the metal ball is speeding up over time. This comes from the influence of gravity, which is represented by the quadratic expression . It is the exponent 2 in that expression that makes it increase by larger and larger amounts.
Another way to study the change in the position of the metal ball is to look at its distance from the ground as a function of time.
Here is a table showing the distance from the ground in feet at 0, 1, 2, and 3 seconds.
| time (seconds) | distance from the ground (feet) |
|---|---|
| 0 | 190 |
| 1 | 174 |
| 2 | 126 |
| 3 | 46 |
Here are those time and distance pairs plotted on a coordinate plane:
Graph of the quadratic function
on a coordinate plane, origin
. Horizontal axis scale 0 to 6 by 1’s, labeled “time (seconds)”. Vertical axis scale 0 to 200 by 50’s, labeled “distance from ground (feet)”. The four points of the function shown are
(0 comma 190), (1 comma 174), (2 comma 126) and (3 comma 46).
The expression that defines the distance from the ground as a function of time is . It tells us that the metal ball's distance from the ground is 190 feet before it is dropped and has decreased by when seconds have passed.