The path of a football flying through the air can be modelled by a quadratic equation. The football reaches the ground after 12 seconds in flight and is kicked from a height of 1 meter. The parabola has undergone a vertical reflection and a vertical compression by a factor of 1/6.
a) Write an equation to represent the path of the football.
b) Does the football reach a height of 8 meters?
Please, if you can, explain in simple language and step by step.
This is probably what they mean:
A parabola is characterized by 3 coefficients, so you need 3 pieces of information to determine a parabola. Two of the pieces of information are given directly, as $f(0) = 1$ and $f(12) = 0$.
For the third, it seems they are attempting to say that a unit parabola was shifted so that the leading coefficient was scaled from $A=1$ to $A=-1/6$. So you have:
$$f(t) = At^2 + Bt + C \tag{1}$$ $$\begin{cases} A=-\frac 16 \\ f(0) = 1 \\ f(12) = 0\end{cases}$$
Can you take it from here?
You know that $A = -1/6$, so (1) becomes:
$$f(t) = -\frac16t^2 + Bt + C \tag{2}$$
Now you know that $f(0) = 1$, so (2) becomes
$$f(0) = -\frac160^2 + B\cdot 0 + C$$ $$1 = 0 + 0 + C$$ $$1 = C$$ So $$f(t) = -\frac16t^2 + Bt + 1\tag{3}$$
Now you just have to find out the value of $B$, so use $f(12) = 0$ (3):
$$f(12) = -\frac16\cdot 12^2 + B\cdot12 + 1$$ $$0 = -\frac{144}{6} + 12 B + 1$$ $$23 = 12 B$$ $$\frac{23}{12} = B$$
So
$$f(t) = -\frac16 t^2 + \frac{23}{12} t + 1$$
Now you know the equation of the height of the football. Then the question becomes, does it ever reach a height of 8m? So set:
$$8 = -\frac16 t^2 + \frac{23}{12} t + 1$$
and solve for $t$ using the quadratic equation. You want to check if there is a positive real number that solves the equation.