Parabolas describe many real things: the height of a thrown ball, the area of a fenced pen, the revenue from a price. The vertex answers “how high” or “how much at most,” and the zeros, found with the quadratic formula or factoring, answer “when” or “where.”
Why it works
Near Earth, gravity pulls a thrown object so that its height is a quadratic in time: $h(t) = -16t^2 + v_0 t + h_0$ in feet, or about $-5t^2 + v_0 t + h_0$ in meters, where $v_0$ is the starting upward speed and $h_0$ the starting height. The negative leading coefficient makes the parabola open down, so the vertex is the highest point. Area problems give quadratics because area multiplies two lengths that both depend on the variable.
Key idea
Write the quadratic, then match the question to a feature: maximum or minimum is the vertex; “when is it zero” or “when does it reach a value” is an equation to solve. Keep only answers that make sense in the story.
Worked examples
Worked example: A launched ball
$h(t) = -5t^2 + 20t + 25$ meters. When does the ball land, and how high does it go?
- Land: $-5t^2 + 20t + 25 = 0$. Divide by $-5$: $t^2 - 4t - 5 = 0$, so $(t - 5)(t + 1) = 0$.
- $t = 5$ or $t = -1$. Negative time is before the launch: it lands at $t = 5$ s.
- Highest point: halfway between the zeros, $t = 2$. $h(2) = -20 + 40 + 25 = 45$ m.
Answer: Lands after $5$ s; maximum height $45$ m.
Worked example: Fencing
With $80$ m of fence along three sides of a pen against a wall, what is the largest area?
- Sides touching the wall: $x$ each; the side opposite the wall: $80 - 2x$. Area $A = x(80 - 2x)$.
- Zeros at $x = 0$ and $x = 40$, so the vertex is at $x = 20$.
- $A(20) = 20 \cdot 40 = 800$.
Answer: $800$ square meters.
Common mistake
Reporting both roots in a time problem, including a negative time. The equation does not know about the story; you do. A launch happens at $t = 0$, so negative times are not part of the answer. Check each root against the situation.