Integrated Math 2 / Unit 3: Quadratic Functions
3.6 Quadratic Models
Notes
Quadratics model projectile motion, area and revenue. The vertex gives the best value.
- Height of a thrown object: $h(t) = -16t^2 + v_0 t + h_0$ in feet, where $v_0$ is initial speed and $h_0$ is starting height.
- The maximum height happens at $\displaystyle t = -\frac{b}{2a}$.
- Example: $h(t) = -16t^2 + 48t + 4$ peaks at $\displaystyle t = \frac{48}{32} = 1.5$ seconds, with $h(1.5) = -36 + 72 + 4 = 40$ feet.
- The constant term is the starting height. A negative or too-large $t$ may not make sense in context.
Practice
For $h(t) = -16t^2 + 48t + 4$, when does the object reach its maximum height?
$\displaystyle t = -\frac{48}{2(-16)} = 1.5$.
For $h(t) = -16t^2 + 48t + 4$, what is the maximum height?
$h(1.5) = -16(2.25) + 72 + 4 = -36 + 72 + 4 = 40$.
A ball is thrown from a roof with $h(t) = -16t^2 + 20t + 6$. How tall is the roof?
At $t = 0$ the height is the constant term, 6 feet.
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