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. Should I enter x = (answer) or just (answer)

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 parabola defined by \( y = ax^2 + bx + c \). The x-coordinate of the vertex is given by the formula:

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

In our equation, \( a = -2 \) and \( b = 6 \). Plugging these values into the vertex formula, we get:

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

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

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

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

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

Now substitute that back into the equation:

\[
y = -2\left(\frac{9}{4}\right) + 6\left(\frac{3}{2}\right)
\]
\[
y = -\frac{18}{4} + \frac{18}{2}
\]
\[
y = -\frac{18}{4} + \frac{36}{4}
\]
\[
y = \frac{36 - 18}{4} = \frac{18}{4} = \frac{9}{2}
\]

Therefore, the maximum value of the parabola is:

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

When asked to enter the answer, if you are asked for the maximum value itself, you should enter \( \frac{9}{2} \).