Curves: When Growth Changes

Lesson 6 of 10 · 7 min · Free, no account needed

What you will learn

When the change itself changes, the graph bends: a parabola.
The highest or lowest point is the vertex. There the change is zero.
A parabola is symmetric: the way down mirrors the way up. Its simplest formula is y = ax² + c.

A ball is thrown straight up at 20 m/s.
Draw its height over time. Your best guess, no wrong answers yet.
Throw it, one second at a time.
Watch the yellow steps: how much higher is the ball after each second?
Find the moment the ball stops going up.
Drag the ball, or the time slider. The blue arrow is its speed. Stop where the arrow disappears, then Check.
This curve has a name.
A curve shaped like this is a parabola. It appears whenever the change itself changes steadily.
Its highest point is the vertex: here (2, 20). At the vertex the change is zero.
A parabola is symmetric: the left side mirrors the right across the line through the vertex.
The simplest parabolas are y = ax² + c. a sets the width and whether it opens up or down; c lifts it.
Shape a fountain jet.
Turn a and c until your curve covers the dashed jet.
Game: catch it.
Each ball flies up and freezes at its top. Tap the ground where it will land. Five throws.
Python can throw the ball too.
Put the three lines in order so the program prints the height each second: 0, 15, 20, 15, 0.
A new throw lands at 6 s. After 1 s it is 25 m high.
Place the pin on the vertex: when is it highest, and how high?
Your first sketch, next to the real throw.
Slide from before to after. What did your hunch get right?

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