Question
Find the maximum value of the parabola: y = -14/5 x^2 + 5. Simplify your answer and write it as a proper fraction, improper fraction, or integer.
Answers
GPT-4o mini
To find the maximum value of the parabola given by the equation:
\[
y = -\frac{14}{5} x^2 + 5
\]
we note that this is a downward-opening parabola because the coefficient of \(x^2\) is negative.
The maximum value of a quadratic function in the form \(y = ax^2 + bx + c\) occurs at the vertex, which can be found using the formula for the x-coordinate of the vertex:
\[
x = -\frac{b}{2a}
\]
For our parabola, \(a = -\frac{14}{5}\) and \(b = 0\) (since there is no \(x\) term). Plugging in these values:
\[
x = -\frac{0}{2 \cdot -\frac{14}{5}} = 0
\]
Next, we substitute \(x = 0\) back into the equation to find the maximum \(y\)-value:
\[
y = -\frac{14}{5}(0)^2 + 5 = 5
\]
Thus, the maximum value of the parabola is:
\[
\boxed{5}
\]
\[
y = -\frac{14}{5} x^2 + 5
\]
we note that this is a downward-opening parabola because the coefficient of \(x^2\) is negative.
The maximum value of a quadratic function in the form \(y = ax^2 + bx + c\) occurs at the vertex, which can be found using the formula for the x-coordinate of the vertex:
\[
x = -\frac{b}{2a}
\]
For our parabola, \(a = -\frac{14}{5}\) and \(b = 0\) (since there is no \(x\) term). Plugging in these values:
\[
x = -\frac{0}{2 \cdot -\frac{14}{5}} = 0
\]
Next, we substitute \(x = 0\) back into the equation to find the maximum \(y\)-value:
\[
y = -\frac{14}{5}(0)^2 + 5 = 5
\]
Thus, the maximum value of the parabola is:
\[
\boxed{5}
\]