Question

Find the equation of the axis of symmetry and the coordinates of the vertex of the graph of the following function.y equals x squared minus 4 x minus 9Question content area bottomPart 1What is the equation of the axis of​ symmetry?enter your response here ​(Type an equation. Simplify your​ answer.)

Answer 1

To find the equation of the axis of symmetry and the coordinates of the vertex for the quadratic function \( y = x^2 - 4x - 9 \), we can use the following formulas:

1. The equation of the axis of symmetry for a quadratic function in the form \( y = ax^2 + bx + c \) is given by:
\[
x = -\frac{b}{2a}
\]
where \( a = 1 \) and \( b = -4 \) in this case.

2. The coordinates of the vertex can be found using the same value of \( x \) obtained from the axis of symmetry and then plugging it back into the function to find the corresponding \( y \) value.

### Step 1: Find the axis of symmetry

Given:
- \( a = 1 \)
- \( b = -4 \)

Calculate \( x \):
\[
x = -\frac{-4}{2 \times 1} = \frac{4}{2} = 2
\]

Thus, the equation of the axis of symmetry is:
\[
\boxed{x = 2}
\]

### Step 2: Find the coordinates of the vertex

Now that we have the \( x \)-coordinate of the vertex, we can find the \( y \)-coordinate by substituting \( x = 2 \) back into the function:
\[
y = (2)^2 - 4(2) - 9
\]
\[
y = 4 - 8 - 9
\]
\[
y = -13
\]

Therefore, the coordinates of the vertex are:
\[
\boxed{(2, -13)}
\]

In conclusion, the equation of the axis of symmetry is \( x = 2 \) and the coordinates of the vertex are \( (2, -13) \).