Step by step
The question
Question: h = −16t² + 80t + 6 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 5 seconds.
t = 0: every part with t is 0.
Only 6 is left: 6.
t = 1: power first.
1 squared is 1.
−16 times 1 is −16.
80 times 1 is 80.
−16 + 80 + 6 = 70.
t = 2: power first.
2 squared is 4.
−16 times 4 is −64.
80 times 2 is 160.
−64 + 160 + 6 = 102.
t = 3: 3 squared is 9.
−16 times 9 is −144.
80 times 3 is 240.
−144 + 240 + 6 = 102.
t = 4: 4 squared is 16.
−16 times 16 is −256.
80 times 4 is 320.
−256 + 320 + 6 = 70.
t = 5: 5 squared is 25.
−16 times 25 is −400.
80 times 5 is 400.
−400 + 400 + 6 = 6.
Look at the heights: 6, 70, 102, 102, 70, 6.
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: 6 at t = 0, and 6 again at t = 5.
The top is halfway between the two times of a pair.
The middle pair is 102 at t = 2 and 102 at t = 3.
Halfway between 2 and 3 is t = 2.5: the top is there, between two columns.
We work out the height there.
2.5 squared is 6.25.
−16 times 6.25 is −100.
80 times 2.5 is 200.
−100 + 200 + 6 = 106.
So the top is at t = 2.5.
There the height is 106 feet.
Careful!
2.5 is the time, in seconds.
The height is h: 106 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 = 2.5, in the middle.
5 Say it in words
So the ball goes up to 106 feet.
It is highest at 2.5 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 80, the number with t.
On top: minus 80.
At the bottom: 2 times −16, that is −32.
−80 divided by −32: minus divided by minus is plus, so 2.5.
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.