Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then
find that value.
-$$f ( x ) = 3 x ^ { 2 } + 2 x - 6$$
A) minimum; $- \frac { 1 } { 3 }$
B) minimum; $- \frac { 19 } { 3 }$
C) maximum; $- \frac { 1 } { 3 }$
D) maximum; $- \frac { 19 } { 3 }$

Question

Quizplus · Accepted Answer

Since the coefficient of $x^2$ is positive (3), the parabola opens upwards, indicating a minimum value. To find the minimum value, use the vertex formula $x = -\frac{b}{2a}$ to find the x-coordinate of the vertex. Here, $a = 3$ and $b = 2$, so $x = -\frac{2}{2*3} = -\frac{1}{3}$. Substitute $x = -\frac{1}{3}$ into the original equation to find the y-coordinate (minimum value): $f(-\frac{1}{3}) = 3(-\frac{1}{3})^2 + 2(-\frac{1}{3}) - 6 = 3(\frac{1}{9}) - \frac{2}{3} - 6 = \frac{1}{3} - \frac{2}{3} - 6 = -\frac{1}{3} - 6 = -\frac{19}{3}$.

Determine, Without Graphing, Whether the Given Quadratic Function Has a Maximum