Question

Name the vertex of the graph of y=x^2 x-27 -120

Answer 1

To find the vertex of the quadratic function \( y = x^2 - 27x - 120 \), we can use the vertex formula for a parabola in the standard form \( y = ax^2 + bx + c \). The x-coordinate of the vertex can be found using:

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

In this case, \( a = 1 \) and \( b = -27 \). Plugging in the values:

\[
x = -\frac{-27}{2 \cdot 1} = \frac{27}{2} = 13.5
\]

Now, we can find the y-coordinate by substituting \( x = 13.5 \) back into the equation:

\[
y = (13.5)^2 - 27(13.5) - 120
\]

Calculating each term:

1. \( (13.5)^2 = 182.25 \)
2. \( -27(13.5) = -364.5 \)

Now substituting these values into the equation:

\[
y = 182.25 - 364.5 - 120
\]

Calculating this step-by-step:

\[
y = 182.25 - 364.5 = -182.25
\]
\[
y = -182.25 - 120 = -302.25
\]

So, the coordinates of the vertex of the graph \( y = x^2 - 27x - 120 \) are:

\[
\left( 13.5, -302.25 \right)
\]

Thus, the vertex is \(\boxed{(13.5, -302.25)}\).