Question

find the y-intercept, the axis of symmetry, and the vertex of the graph of the function.\\( f(x) = -x^2 - 2x - 9 \\)\\( \\)\\( \\)the y-intercept is \\( (0, -9) \\). (type an ordered pair.)\\( \\)the axis of symmetry is \\( \square \\). (simplify your answer. type an equation.)

Explanation:

Step1: Recall the formula for the axis of symmetry of a quadratic function.

For a quadratic function in the form \( f(x) = ax^2 + bx + c \), the axis of symmetry is given by the equation \( x = -\frac{b}{2a} \).

Step2: Identify the values of \( a \) and \( b \) from the given function.

In the function \( f(x) = -x^2 - 2x - 9 \), we have \( a = -1 \) and \( b = -2 \).

Step3: Substitute the values of \( a \) and \( b \) into the formula.

Substitute \( a = -1 \) and \( b = -2 \) into \( x = -\frac{b}{2a} \):
\[
x = -\frac{-2}{2 \times (-1)}
\]

Step4: Simplify the expression.

First, simplify the numerator: \( -(-2) = 2 \).
Then, simplify the denominator: \( 2 \times (-1) = -2 \).
So, \( x = \frac{2}{-2} = -1 \).

Answer:

The axis of symmetry is \( x = -1 \).