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.
-$$f ( x ) = - 4 x ^ { 2 } + 4 x$$
A) maximum; $\left( \frac { 1 } { 2 } , 1 \right)$
B) minimum; $\left( \frac { 1 } { 2 } , 1 \right)$
C) minimum; $\left( - \frac { 1 } { 2 } , - 1 \right)$
D) maximum; $\left( - \frac { 1 } { 2 } , - 1 \right)$

Question

Quizplus · Accepted Answer

The quadratic function $f(x) = -4x^2 + 4x$ has a negative leading coefficient (-4), indicating that it opens downwards and thus has a maximum value. To find the coordinates of the maximum point, we use the formula for the vertex of a quadratic function, $x = -\frac{b}{2a}$, where $a = -4$ and $b = 4$. Substituting these values gives $x = -\frac{4}{2(-4)} = \frac{1}{2}$. Substituting $x = \frac{1}{2}$ back into the function gives $f\left(\frac{1}{2}\right) = -4\left(\frac{1}{2}\right)^2 + 4\left(\frac{1}{2}\right) = -1 + 2 = 1$. Therefore, the maximum point is $\left(\frac{1}{2}, 1\right)$.

Determine a Quadratic Function's Minimum or Maximum Value Determine Whether f(x)=−4x2+4xf ( x ) = - 4 x ^ { 2 } + 4 xf(x)=−4x2+4x

Determine a Quadratic Function's Minimum or Maximum Value
Determine Whether $f ( x ) = - 4 x ^ { 2 } + 4 x$