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

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