Step by step
The question
Question: h = −16t² + 64t + 5 is the height, in feet, of a ball t seconds after it is thrown.
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 thrown.
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 4 seconds.
t = 0: every part with t is 0.
Only 5 is left: 5.
t = 1: power first.
1 squared is 1.
−16 times 1 is −16.
64 times 1 is 64.
−16 + 64 + 5 = 53.
t = 2: power first.
2 squared is 4.
−16 times 4 is −64.
64 times 2 is 128.
−64 + 128 + 5 = 69.
t = 3: 3 squared is 9.
−16 times 9 is −144.
64 times 3 is 192.
−144 + 192 + 5 = 53.
t = 4: 4 squared is 16.
−16 times 16 is −256.
64 times 4 is 256.
−256 + 256 + 5 = 5.
Look at the heights: 5, 53, 69, 53, 5.
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: 5 at t = 0, and 5 again at t = 4.
The top is halfway between the two times of a pair.
The top is in the middle of each pair: t = 2.
There the height is the biggest in the table: 69 feet.
Careful!
2 is the time, in seconds.
The height is h: 69 feet.
4 Draw the flight
We draw the flight: time t across, height h up.
The h numbers go up in 10s.
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 = 2, in the middle.
5 Say it in words
So the ball goes up to 69 feet.
It is highest at 2 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 64, the number with t.
On top: minus 64.
At the bottom: 2 times −16, that is −32.
−64 divided by −32: minus divided by minus is plus, so 2.
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.