College Algebra, Chapter 4, 4.1, Section 4.1, Problem 44

Determine a function whose graph is a parabola with vertex $(3, 4)$ and that passes through the point $(1, -8)$.

Recall that the general equation of a parabola is

$f(x) = a(x - h)^2 + k$, with vertex $(h,k)$

where $x = 1, f(x) = -8, h = 3$ and $k = 4$.

Solving for $a$, we have


$\begin{equation} \begin{aligned} -8 =& a(1 - 3)^2 + 4 && \text{Substitute the given values} \\ \\ -8 =& 4a + 4 && \text{Evaluate the parentheses} \\ \\ -8 - 4 =& 4a && \text{Subtract } 4 \\ \\ -12 =& 4a && \text{Divide by } 4 \\ \\ a =& -3 && \text{Answer} \end{aligned} \end{equation}$


Thus, the function is

$f(x) = -3 (x - 3)^2 + 4$