Question

Find the maximum value of the parabola: y = -2x^2 + 6x. Simplify your answer and write it as a proper fraction, improper fraction, or integer.

Answer 1

To find the maximum value of the parabola given by the equation \( y = -2x^2 + 6x \), we can use the vertex formula for a quadratic function of the form \( y = ax^2 + bx + c \). Here, \( a = -2 \) and \( b = 6 \).

The x-coordinate of the vertex can be calculated using the formula:

\[
x = -\frac{b}{2a}
\]

Substituting the values of \( a \) and \( b \):

\[
x = -\frac{6}{2 \cdot -2} = -\frac{6}{-4} = \frac{3}{2}
\]

Next, we substitute \( x = \frac{3}{2} \) back into the original equation to find the corresponding y-coordinate:

\[
y = -2\left(\frac{3}{2}\right)^2 + 6\left(\frac{3}{2}\right)
\]

Calculating \( \left(\frac{3}{2}\right)^2 \):

\[
\left(\frac{3}{2}\right)^2 = \frac{9}{4}
\]

Now we can substitute this back into the equation:

\[
y = -2\left(\frac{9}{4}\right) + 6\left(\frac{3}{2}\right)
\]

Calculating \( -2 \cdot \frac{9}{4} \):

\[
-2 \cdot \frac{9}{4} = -\frac{18}{4} = -\frac{9}{2}
\]

Calculating \( 6 \cdot \frac{3}{2} \):

\[
6 \cdot \frac{3}{2} = \frac{18}{2} = 9
\]

Now we can find \( y \):

\[
y = -\frac{9}{2} + 9 = -\frac{9}{2} + \frac{18}{2} = \frac{9}{2}
\]

Thus, the maximum value of the parabola is:

\[
\boxed{\frac{9}{2}}
\]