Answer
(a.) $32$, The driver's height is $32$ feet after $1$ second.
(b.) $0$, The driver hit the water after $2$ seconds.
(c.) $ f(t)=-16(t-2)(t+1)$.
(d.) $f(1)=32$ and $f(2)=0$.
Work Step by Step
The given function is
$\Rightarrow f(t)=-16t^2+16t+32$
(a.) Replace $x$ with $1$ into the function.
$\Rightarrow f(1)=-16(1)^2+16(1)+32$
Simplify.
$\Rightarrow f(1)=-16+16+32$
Add like terms.
$\Rightarrow f(1)=32$
The driver's height is $32$ feet after $1$ second.
(b.) Replace $x$ with $2$ in the function.
$\Rightarrow f(2)=-16(2)^2+16(2)+32$
Simplify.
$\Rightarrow f(2)=-64+32+32$
Add like terms.
$\Rightarrow f(2)=0$
The driver hit the water after $2$ seconds.
(c.) Factor the function.
$\Rightarrow f(t)=-16t^2+16t+32$
Factor out $-16$ from all term.
$\Rightarrow f(t)=-16(t^2-t-2)$
Rewrite the middle term $-t$ as $-2t+t$.
$\Rightarrow f(t)=-16(t^2-2t+t-2)$
Group terms.
$\Rightarrow f(t)=-16[(t^2-2t)+(t-2)]$
Factor each group.
$\Rightarrow f(t)=-16[t(t-2)+1(t-2)]$
Factor out $(t-2)$.
$\Rightarrow f(t)=-16(t-2)(t+1)$.
(d.) Replace $x$ with $1$ into factor form.
$\Rightarrow f(1)=-16(1-2)(1+1)$
Add like terms.
$\Rightarrow f(1)=-16(-1)(2)$
Simplify.
$\Rightarrow f(1)=32$
Replace $x$ with $2$ in factor form.
$\Rightarrow f(2)=-16(2-2)(2+1)$
Add like terms.
$\Rightarrow f(2)=-16(0)(3)$
Simplify.
$\Rightarrow f(2)=0$.
