Worked examples
The question
Question: h = −16t² + 64t + 5 is the height, in feet, of a ball t seconds after it is thrown.
What does 5 mean?
We write the equation big, so we can label it.
h = −16t2 + 64t + 5
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 = 0: −16 × 02 + 64 × 0 + 5 = 5
5
t = 1: power first.
1 squared is 1.
−16 times 1 is −16.
64 times 1 is 64.
−16 + 64 + 5 = 53.
t = 1: −16 × 12 + 64 × 1 + 5 = −16 + 64 + 5 = 53
53
t = 2: power first.
2 squared is 4.
−16 times 4 is −64.
64 times 2 is 128.
−64 + 128 + 5 = 69.
t = 2: −16 × 22 + 64 × 2 + 5 = −64 + 128 + 5 = 69
69
t = 3: 3 squared is 9.
−16 times 9 is −144.
64 times 3 is 192.
−144 + 192 + 5 = 53.
t = 3: −16 × 32 + 64 × 3 + 5 = −144 + 192 + 5 = 53
53
t = 4: 4 squared is 16.
−16 times 16 is −256.
64 times 4 is 256.
−256 + 256 + 5 = 5.
t = 4: −16 × 42 + 64 × 4 + 5 = −256 + 256 + 5 = 5
5
Look at the heights: 5, 53, 69, 53, 5.
They go up, then come down, like the ball.
3 Find the numbers in the table
Now we write the equation again, and label its numbers.
h = −16t2 + 64t + 5
At t = 0, every part with t is 0.
Only 5 is left: the first height in the table.
So 5 is the height when the ball is thrown.
Careful!
5 is only the height at t = 0.
The table goes up to 69 feet, so the top is higher.
Careful: not top = 5 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 curve starts at 5 feet, not at 0: the ball starts 5 feet up.
5 Say it in words
So 5 means: the ball is 5 feet high when it is thrown, at t = 0.
6 Check
Let's check: put t = 0 in.
Every part with t is 0, so we get 5.
It works!
t = 0: −16 × 02 + 64 × 0 + 5 = 0 + 0 + 5 = 5
Let's look back.
First: what the letters mean.
Then: make a small table.
Then: find the numbers in the table.
Then: draw the flight.
Then: say it in words.
Last: check.
Answer h = −16t2 + 64t + 5
▶ Watch it solved
The question
Question: h = −16t² + 48t + 4 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.
h = −16t2 + 48t + 4
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 = 0: −16 × 02 + 48 × 0 + 4 = 4
4
t = 1: power first.
1 squared is 1.
−16 times 1 is −16.
48 times 1 is 48.
−16 + 48 + 4 = 36.
t = 1: −16 × 12 + 48 × 1 + 4 = −16 + 48 + 4 = 36
36
t = 2: power first.
2 squared is 4.
−16 times 4 is −64.
48 times 2 is 96.
−64 + 96 + 4 = 36.
t = 2: −16 × 22 + 48 × 2 + 4 = −64 + 96 + 4 = 36
36
t = 3: 3 squared is 9.
−16 times 9 is −144.
48 times 3 is 144.
−144 + 144 + 4 = 4.
t = 3: −16 × 32 + 48 × 3 + 4 = −144 + 144 + 4 = 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.
t = 1.5: −16 × 1.52 + 48 × 1.5 + 4 = −36 + 72 + 4 = 40
So the top is at t = 1.5.
t = 1.5
There the height is 40 feet.
h = 40
Careful!
1.5 is the time, in seconds.
The height is h: 40 feet.
Careful: not 1.5 feet high ✗
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 goes up to 40 feet.
It is highest at 1.5 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 48, the number with t.
a = −16, b = 48
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!
t = −482 × (−16) = −48−32 = 1.5
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.
▶ Watch it solved
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.
h = −16t2 + 80t + 6
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 = 0: −16 × 02 + 80 × 0 + 6 = 6
6
t = 1: power first.
1 squared is 1.
−16 times 1 is −16.
80 times 1 is 80.
−16 + 80 + 6 = 70.
t = 1: −16 × 12 + 80 × 1 + 6 = −16 + 80 + 6 = 70
70
t = 2: power first.
2 squared is 4.
−16 times 4 is −64.
80 times 2 is 160.
−64 + 160 + 6 = 102.
t = 2: −16 × 22 + 80 × 2 + 6 = −64 + 160 + 6 = 102
102
t = 3: 3 squared is 9.
−16 times 9 is −144.
80 times 3 is 240.
−144 + 240 + 6 = 102.
t = 3: −16 × 32 + 80 × 3 + 6 = −144 + 240 + 6 = 102
102
t = 4: 4 squared is 16.
−16 times 16 is −256.
80 times 4 is 320.
−256 + 320 + 6 = 70.
t = 4: −16 × 42 + 80 × 4 + 6 = −256 + 320 + 6 = 70
70
t = 5: 5 squared is 25.
−16 times 25 is −400.
80 times 5 is 400.
−400 + 400 + 6 = 6.
t = 5: −16 × 52 + 80 × 5 + 6 = −400 + 400 + 6 = 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.
t = 2.5: −16 × 2.52 + 80 × 2.5 + 6 = −100 + 200 + 6 = 106
So the top is at t = 2.5.
t = 2.5
There the height is 106 feet.
h = 106
Careful!
2.5 is the time, in seconds.
The height is h: 106 feet.
Careful: not 2.5 feet high ✗
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.
a = −16, b = 80
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!
t = −802 × (−16) = −80−32 = 2.5
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.
▶ Watch it solved
