Question

A. find an equation for f(t) B. explain how you can use the graph's t-intercept to check the reasonableness of your equation.

Answer 1

the form of a parabola has the form

$f(t)=at^2+bt+c$

when t=0

$f(0)=c$

usinc the graphic when t = 0 the value on the y axis is 45

$c=45$

now using the formula for the vertex:

$\begin{gathered} \text{vertex}=(h,k) \\ h=(-\frac{b}{2a});k=f(-\frac{b}{2a}) \\ \end{gathered}$

since h=0

$0=-\frac{b}{2a}\rightarrow b=0$

the only possible way for this to be 0 is if the numerator is equal to 0, reason why b is 0

now using the point given, we find a

$\begin{gathered} y=at^2+c \\ 29=a+45 \\ 29-45=a \\ a=-16 \end{gathered}$

re write the equation

$f(t)=-16x^2+45$

there is not h, inside the parentheses beacuse h=0

Does the equation makes sense?

Yes, because the number accompanying the x^2 the parabola is upside-down, also the vertex its on (0,45) menaing the parabola moved 45 units up.

Also since the hammer is dropping the y should become less ultil it gets to the floor.