Zeros: Where the Parabola Lands

Lesson 9 of 12 · 7 min · Free, no account needed

What you will learn

A zero of a function is an x where its value is 0: where the graph meets the x axis.
Zero product: a product is 0 only when one of its factors is 0.
So y = (x − p)(x − q) lands at x = p and x = q. Factor first, then read the zeros.

A ball is thrown up at 20 m/s. When does it hit the ground?
You see the first second of its flight. Drag the flag to your bet.
20t − 5t² is the same as 5 × t × (4 − t).
Turn time. The height is the area of a t by (4 − t) rectangle, times 5. Find both moments it's on the ground.
Switch the factors off.
Tap each bracket. A bracket is off when it equals 0.
Zeros and the zero product.
A zero of a function is an input that gives 0. On the graph, it's where the curve meets the x axis.
Zero product: if A × B = 0, then A = 0 or B = 0. Nothing else multiplies to 0.
So h = 5t(4 − t) is 0 at t = 0 and t = 4. Factored form shows the zeros at a glance.
Pin down y = (x + 1)(x − 3).
Drag the two pins along the x axis. The curve passes through them.
Game: landing pads.
Each rocket flies along its formula. Drag the pad to where it lands, then Launch. Factor the last two first.
Design a jump of your own.
Fill the two blanks: where your jump takes off and where it lands.
A shop's revenue is R = p(20 − p) at price p. What price earns the most?
At $0 and at $20 it earns nothing. The top of a parabola sits halfway between its zeros.
A frog tries your jump.
Press Step to hop. Watch the two factors at each hop.

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