Find the range of the given function.
-$$f ( x ) = 3 x ^ { 2 } + 24 x + 51$$
A) $[ 3 , \infty )$
B) $( - \infty , 4 ]$
C) $( - \infty , - 3 ]$
D) $[ - 4 , \infty )$

Question

Quizplus · Accepted Answer

The function $f(x) = 3x^2 + 24x + 51$ is a quadratic function, which opens upwards (since the coefficient of $x^2$ is positive). To find the range, we need to find the vertex of the parabola, as it represents the minimum point of the function. The x-coordinate of the vertex is given by $-\frac{b}{2a}$, where $a = 3$ and $b = 24$, so $-\frac{24}{2*3} = -4$. Plugging this back into the function gives the minimum value: $f(-4) = 3(-4)^2 + 24(-4) + 51 = 48 - 96 + 51 = 3$. Therefore, the range is $[3, \infty)$.

Find the Range of the Given Function f(x)=3x2+24x+51f ( x ) = 3 x ^ { 2 } + 24 x + 51f(x)=3x2+24x+51

Find the Range of the Given Function $f ( x ) = 3 x ^ { 2 } + 24 x + 51$