Step by step
The question
Question: h = −5t² + 20t + 3 is the height, in meters, of a ball t seconds after it is kicked.
What does 3 mean?
We write the equation big, so we can label it.
1 What the letters mean
h is the height of the ball, in meters.
t is the time, in seconds, after it is kicked.
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 3 is left: 3.
t = 1: power first.
1 squared is 1.
−5 times 1 is −5.
20 times 1 is 20.
−5 + 20 + 3 = 18.
t = 2: power first.
2 squared is 4.
−5 times 4 is −20.
20 times 2 is 40.
−20 + 40 + 3 = 23.
t = 3: 3 squared is 9.
−5 times 9 is −45.
20 times 3 is 60.
−45 + 60 + 3 = 18.
t = 4: 4 squared is 16.
−5 times 16 is −80.
20 times 4 is 80.
−80 + 80 + 3 = 3.
Look at the heights: 3, 18, 23, 18, 3.
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.
At t = 0, every part with t is 0.
Only 3 is left: the first height in the table.
So 3 is the height when the ball is kicked.
Careful!
3 is only the height at t = 0.
The table goes up to 23 meters, so the top is higher.
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 curve starts at 3 meters, not at 0: the ball starts 3 meters up.
5 Say it in words
So 3 means: the ball is 3 meters high when it is kicked, at t = 0.
6 Check
Let's check: put t = 0 in.
Every part with t is 0, so we get 3.
It works!
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.