Determine a Quadratic Function's Minimum or Maximum Value
Determine whether the given quadratic function has a minimum value or maximum value. Then find the
coordinates of the minimum or maximum point.
-$$f ( x ) = 3 x ^ { 2 } + 2 x - 2$$
A) minimum; $\left( - \frac { 1 } { 3 } , - \frac { 7 } { 3 } \right)$
B) maximum; $\left( - \frac { 1 } { 3 } , - \frac { 7 } { 3 } \right)$
C) minimum; $\left( - \frac { 7 } { 3 } , - \frac { 1 } { 3 } \right)$
D) maximum; $\left( - \frac { 7 } { 3 } , - \frac { 1 } { 3 } \right)$

Question

Quizplus · Accepted Answer

We can determine whether the quadratic function has a minimum or maximum by looking at the coefficient of the squared term. In this case, the coefficient is positive (3), so the graph opens upwards and has a minimum value. To find the coordinates of the minimum point, we use the formula $-\frac{b}{2a}$ to find the x-coordinate and plug that value into the function to find the corresponding y-coordinate.

For this function, we have $a=3$ and $b=2$. Using the formula, we get:
$$x=-\frac{b}{2a}=-\frac{2}{2(3)}=-\frac{1}{3}$$
To find the y-coordinate, we plug this value back into the function:
$$f\left(-\frac{1}{3}\right)=3\left(-\frac{1}{3}\right)^2+2\left(-\frac{1}{3}\right)-2=-\frac{7}{3}$$
Therefore, the minimum point is $\left(-\frac{1}{3},-\frac{7}{3}\right)$, and the answer is A.

Determine a Quadratic Function's Minimum or Maximum Value Determine Whether f(x)=3x2+2x−2f ( x ) = 3 x ^ { 2 } + 2 x - 2f(x)=3x2+2x−2

Determine a Quadratic Function's Minimum or Maximum Value
Determine Whether $f ( x ) = 3 x ^ { 2 } + 2 x - 2$