Step by step
The question
Question: h = −16t² + 96t + 3 is the height, in feet, of a ball t seconds after it is hit.
How high does the ball go?
We write the equation big, so we can label it.
1 What the letters mean
h is the height of the ball, in feet.
t is the time, in seconds, after it is hit.
2 Make a small table
To see the flight, we try t = 0, 1, 2 and so on, until the ball is back at its first height.
That is 6 seconds.
t = 0: every part with t is 0.
Only 3 is left: 3.
t = 1: power first.
1 squared is 1.
−16 times 1 is −16.
96 times 1 is 96.
−16 + 96 + 3 = 83.
t = 2: power first.
2 squared is 4.
−16 times 4 is −64.
96 times 2 is 192.
−64 + 192 + 3 = 131.
t = 3: 3 squared is 9.
−16 times 9 is −144.
96 times 3 is 288.
−144 + 288 + 3 = 147.
t = 4: 4 squared is 16.
−16 times 16 is −256.
96 times 4 is 384.
−256 + 384 + 3 = 131.
t = 5: 5 squared is 25.
−16 times 25 is −400.
96 times 5 is 480.
−400 + 480 + 3 = 83.
t = 6: 6 squared is 36.
−16 times 36 is −576.
96 times 6 is 576.
−576 + 576 + 3 = 3.
Look at the heights: 3, 83, 131, 147, 131, 83, 3.
They go up, then come down, like the ball.
3 Find the highest point
The ball goes up and comes down the same way.
So it passes each height twice: once going up, once coming down.
The heights come in pairs: 3 at t = 0, and 3 again at t = 6.
The top is halfway between the two times of a pair.
The top is in the middle of each pair: t = 3.
There the height is the biggest in the table: 147 feet.
Careful!
3 is the time, in seconds.
The height is h: 147 feet.
4 Draw the flight
We draw the flight: time t across, height h up.
The h numbers go up in 20s.
Put the points from the table on the grid.
Join them with a smooth curve: the path of the height over time.
The top of the curve is at t = 3, in the middle.
5 Say it in words
So the ball goes up to 147 feet.
It is highest at 3 seconds.
6 Check
Let's check with the rule from the parabola lesson.
It finds the same halfway point with no table: the top is at t = minus b/2a.
Here a is −16, the number with t squared.
b is 96, the number with t.
On top: minus 96.
At the bottom: 2 times −16, that is −32.
−96 divided by −32: minus divided by minus is plus, so 3.
The same top.
It works!
Let's look back.
First: what the letters mean.
Then: make a small table.
Then: find the highest point.
Then: draw the flight.
Then: say it in words.
Last: check.