Question

Find the vertex of the parabola: y = x^2 + 5x. Simplify both coordinates and write them as proper fractions, improper fractions, or integers.

Answer 1

To find the vertex of the parabola given by the equation \( y = x^2 + 5x \), we can use the vertex formula for a parabola in the form \( y = ax^2 + bx + c \). The x-coordinate of the vertex can be found using the formula:

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

Here, \( a = 1 \) and \( b = 5 \).

Substituting these values into the formula:

\[
x = -\frac{5}{2 \cdot 1} = -\frac{5}{2}
\]

Next, we need to find the corresponding y-coordinate by substituting \( x = -\frac{5}{2} \) back into the original equation:

\[
y = \left(-\frac{5}{2}\right)^2 + 5\left(-\frac{5}{2}\right)
\]

Calculating \( \left(-\frac{5}{2}\right)^2 \):

\[
\left(-\frac{5}{2}\right)^2 = \frac{25}{4}
\]

Now, calculating \( 5\left(-\frac{5}{2}\right) \):

\[
5\left(-\frac{5}{2}\right) = -\frac{25}{2}
\]

We need to combine \( \frac{25}{4} \) and \( -\frac{25}{2} \). To do this, we convert \( -\frac{25}{2} \) to have a common denominator with \( \frac{25}{4} \):

\[
-\frac{25}{2} = -\frac{25 \cdot 2}{2 \cdot 2} = -\frac{50}{4}
\]

Now we can add:

\[
y = \frac{25}{4} - \frac{50}{4} = \frac{25 - 50}{4} = \frac{-25}{4}
\]

Thus, the vertex of the parabola is:

\[
\left(-\frac{5}{2}, -\frac{25}{4}\right)
\]

So, the final answer is:

\[
\text{Vertex: } \left(-\frac{5}{2}, -\frac{25}{4}\right)
\]