Find the range of the quadratic function in interval notation.
-$$f ( x ) = - x ^ { 2 } + 14 x + 2$$
A) $[ 7 , \infty )$
B) $( - \infty , 7 ]$
C) $( - \infty , 51 ]$
D) $[ 51 , \infty )$

Question

Quizplus · Accepted Answer

The vertex form of a quadratic function $f(x) = ax^2 + bx + c$ is $f(x) = a(x-h)^2 + k$, where $(h, k)$ is the vertex of the parabola. The given function is $f(x) = -x^2 + 14x + 2$. To find the vertex, we use the formula $h = -\frac{b}{2a}$. Here, $a = -1$ and $b = 14$, so $h = -\frac{14}{2(-1)} = 7$. To find $k$, we substitute $h$ back into the original equation: $k = -(7)^2 + 14(7) + 2 = -49 + 98 + 2 = 51$. Since $a = -1 < 0$, the parabola opens downwards, making the vertex the maximum point. Therefore, the range is $(-\infty, 51]$.

Find the Range of the Quadratic Function in Interval Notation f(x)=−x2+14x+2f ( x ) = - x ^ { 2 } + 14 x + 2f(x)=−x2+14x+2

Find the Range of the Quadratic Function in Interval Notation $f ( x ) = - x ^ { 2 } + 14 x + 2$