The graph of the quadratic $y = ax^2 + bx + c$ has the following properties: (1) The maximum value of $y = ax^2 + bx + c$ is 5, which occurs at $x = 3$. (2) The graph passes through the point $(0,-13)$. If the graph passes through the point $(4,m)$, then what is the value of $m$?
I need to find the value of $y$ when $x = 4$, so I need to know the coefficients $a, b, c$ of the quadratic. Since the graph has a maximum at $x = 3$, I know that the axis of symmetry is $x = 3$, which means that the vertex form of the quadratic is $y = a(x - 3)^2 + 5$. To find $a$, I can use the fact that the graph passes through $(0,-13)$. Plugging in $x = 0$ and $y = -13$, I get $-13 = a(0 - 3)^2 + 5$, which simplifies to $-18 = 9a$, so $a = -2$. Now I have $y = -2(x - 3)^2 + 5$. To find $m$, I can plug in $x = 4$ and solve for $y$. I get $y = -2(4 - 3)^2 + 5$, which simplifies to $y = -2 + 5$, so $y = 3$. Therefore, $m = 3$. 