What is the $y$-value of the vertex of the function $f(x)=-(x+8)(x-14)$ ?
The given function is \(f(x) = -(x + 8)(x - 14)\). This is a quadratic function provided in its factored form, which is generally written as \(f(x) = a(x - r_1)(x - r_2)\). By comparing our function to this form, we can identify the leading coefficient \(a = -1\) and the roots of the function (the points where \(f(x) = 0\)), which are \(r_1 = -8\) and \(r_2 = 14\). The graph of this function is a downward-opening parabola, and its vertex represents the maximum point on the curve.
The \(x\)-coordinate of the vertex, denoted as \(h\), lies exactly halfway between the two roots of the quadratic function. We calculate this by finding the average of the roots:
\[h = \frac{r_1 + r_2}{2} = \frac{-8 + 14}{2} = \frac{6}{2} = 3.\]
Thus, the vertex of the parabola occurs at \(x = 3\).
To find the \(y\)-coordinate of the vertex, \(k\), we substitute the \(x\)-coordinate \(h = 3\) into the original function \(f(x)\):
\[k = f(3) = -(3 + 8)(3 - 14).\]
Calculating the terms inside the parentheses gives us:
\[k = -(11)(-11).\]
Performing the multiplication:
\[k = -(-121) = 121.\]
Therefore, the \(y\)-coordinate of the vertex is \(121\).
Comparing our calculated value of \(121\) to the provided options (3, 6, 112, 121), we see that our result matches the fourth option. The \(y\)-coordinate of the vertex of the function \(f(x) = -(x + 8)(x - 14)\) is \(\boxed{121}\).
The concept of finding the vertex of a parabola is essential in physics and engineering. For instance, when a projectile is launched into the air, its height over time follows a parabolic path. By identifying the vertex of the quadratic equation that models the trajectory, engineers can determine the maximum height the object will reach, ensuring safety and precision in applications like designing sports equipment or calculating flight paths.