h = -5t^2 + 30t + 1 is the height, in meters, of a rocket t seconds… Step by Step

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The question

Question: h = −5t² + 30t + 1 is the height, in meters, of a rocket t seconds after it is launched.

How high does the rocket go?

We write the equation big, so we can label it.

h = −5t² + 30t + 1

1 What the letters mean

h is the height of the rocket, in meters.

t is the time, in seconds, after it is launched.

2 Make a small table

To see the flight, we try t = 0, 1, 2 and so on, until the rocket is back at its first height.

That is 6 seconds.

t = 0: every part with t is 0.

Only 1 is left: 1.

t = 0: −5 × 0² + 30 × 0 + 1 = 1

t = 1: power first.

1 squared is 1.

−5 times 1 is −5.

30 times 1 is 30.

−5 + 30 + 1 = 26.

t = 1: −5 × 1² + 30 × 1 + 1 = −5 + 30 + 1 = 26

t = 2: power first.

2 squared is 4.

−5 times 4 is −20.

30 times 2 is 60.

−20 + 60 + 1 = 41.

t = 2: −5 × 2² + 30 × 2 + 1 = −20 + 60 + 1 = 41

t = 3: 3 squared is 9.

−5 times 9 is −45.

30 times 3 is 90.

−45 + 90 + 1 = 46.

t = 3: −5 × 3² + 30 × 3 + 1 = −45 + 90 + 1 = 46

t = 4: 4 squared is 16.

−5 times 16 is −80.

30 times 4 is 120.

−80 + 120 + 1 = 41.

t = 4: −5 × 4² + 30 × 4 + 1 = −80 + 120 + 1 = 41

t = 5: 5 squared is 25.

−5 times 25 is −125.

30 times 5 is 150.

−125 + 150 + 1 = 26.

t = 5: −5 × 5² + 30 × 5 + 1 = −125 + 150 + 1 = 26

t = 6: 6 squared is 36.

−5 times 36 is −180.

30 times 6 is 180.

−180 + 180 + 1 = 1.

t = 6: −5 × 6² + 30 × 6 + 1 = −180 + 180 + 1 = 1

Look at the heights: 1, 26, 41, 46, 41, 26, 1.

They go up, then come down, like the rocket.

3 Find the highest point

The rocket 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: 1 at t = 0, and 1 again at t = 6.

The top is halfway between the two times of a pair.

The top is in the middle of each pair: t = 3.

t = 3

There the height is the biggest in the table: 46 meters.

h = 46

Careful!

3 is the time, in seconds.

The height is h: 46 meters.

Careful: not 3 meters high ✗

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 = 3, in the middle.

5 Say it in words

So the rocket goes up to 46 meters.

It is highest at 3 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 −5, the number with t squared.

b is 30, the number with t.

a = −5, b = 30

On top: minus 30.

At the bottom: 2 times −5, that is −10.

−30 divided by −10: minus divided by minus is plus, so 3.

The same top.

It works!

t = −302 × (−5) = −30−10 = 3

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.

Answer