Question

the height h, in feet, of a coin being tossed at a football game is given by the function $h = -16t^2 + 30t + 4$, where t is the time in seconds. sketch the graph of this function. when will the coin hit the ground?

Explanation:

Step1: Identify target condition

Set $h=0$ (height at ground)
$$-16t^2 + 30t + 4 = 0$$

Step2: Simplify the quadratic equation

Multiply by $-1$:
$$16t^2 - 30t - 4 = 0$$
Divide by 2:
$$8t^2 - 15t - 2 = 0$$

Step3: Factor the quadratic

Find factors of $8*(-2)=-16$ that sum to $-15$: $-16$ and $1$
$$8t^2 -16t + t -2 = 0$$
$$8t(t-2) +1(t-2)=0$$
$$(8t+1)(t-2)=0$$

Step4: Solve for valid t

Time cannot be negative, so discard $t=-\frac{1}{8}$
$$t=2$$

Step5: Verify graph shape

The quadratic $h=-16t^2+30t+4$ has a negative leading coefficient, so it opens downward. Initial height at $t=0$ is $h=4$, which matches the second (top-middle) graph (starts at $h=4$, peaks, then falls to $t=2, h=0$).

Answer:

The correct graph is the top-middle one (opening downward, starting at $h=4$, hitting $h=0$ at $t=2$). The coin hits the ground at $t=2$ seconds.