Step by step
The question
Question: h = −16t² + 48t + 4 is the height, in feet, of a ball t seconds after it is thrown.
When is the ball highest?
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 3 seconds.
t = 0: every part with t is 0.
Only 4 is left: 4.
t = 1: power first.
1 squared is 1.
−16 times 1 is −16.
48 times 1 is 48.
−16 + 48 + 4 = 36.
t = 2: power first.
2 squared is 4.
−16 times 4 is −64.
48 times 2 is 96.
−64 + 96 + 4 = 36.
t = 3: 3 squared is 9.
−16 times 9 is −144.
48 times 3 is 144.
−144 + 144 + 4 = 4.
Look at the heights: 4, 36, 36, 4.
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: 4 at t = 0, and 4 again at t = 3.
The top is halfway between the two times of a pair.
The middle pair is 36 at t = 1 and 36 at t = 2.
Halfway between 1 and 2 is t = 1.5: the top is there, between two columns.
We work out the height there.
1.5 squared is 2.25.
−16 times 2.25 is −36.
48 times 1.5 is 72.
−36 + 72 + 4 = 40.
So the top is at t = 1.5.
There the height is 40 feet.
Careful!
40 is the height, in feet.
The time is t: 1.5 seconds.
4 Draw the flight
We draw the flight: time t across, height h up.
The h numbers go up in 5s.
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 = 1.5, in the middle.
5 Say it in words
So the ball is highest at 1.5 seconds.
Then it is 40 feet high.
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 48, the number with t.
On top: minus 48.
At the bottom: 2 times −16, that is −32.
−48 divided by −32: minus divided by minus is plus, so 1.5.
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.