Questions: For the quadratic function f(x)=-x^2-8x, answer parts (a) through (c). finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any. Does the graph of f open up or down? What are the coordinates of the vertex? What is the equation of the axis of symmetry? What is/are the x-intercept(s)? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The x-intercept(s) is/are B. There are no x-intercepts.

Solution Steps

To solve the given quadratic function \( f(x) = -x^2 - 8x \), we need to find the vertex, axis of symmetry, and x-intercepts.

1. Vertex: The vertex of a quadratic function \( ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \). Plug this x-value back into the function to get the y-coordinate.
2. Axis of Symmetry: The axis of symmetry is the vertical line that passes through the vertex, given by \( x = -\frac{b}{2a} \).
3. X-Intercepts: To find the x-intercepts, set \( f(x) = 0 \) and solve the quadratic equation \( -x^2 - 8x = 0 \).

To find the vertex of the quadratic function \( f(x) = -x^2 - 8x \), we use the formula for the x-coordinate of the vertex:

\[ x = -\frac{b}{2a} = -\frac{-8}{2 \cdot -1} = -4 \]

Next, we substitute \( x = -4 \) back into the function to find the y-coordinate:

\[ y = f(-4) = -(-4)^2 - 8(-4) = -16 + 32 = 16 \]

Thus, the vertex is

\[ \boxed{(-4, 16)} \]

The axis of symmetry for a quadratic function is given by the x-coordinate of the vertex. Therefore, the equation of the axis of symmetry is:

\[ x = -4 \]

This can be expressed as

\[ \boxed{x = -4} \]

To find the x-intercepts, we set the function equal to zero:

\[ -x^2 - 8x = 0 \]

Factoring out \(-x\):

\[ -x(x + 8) = 0 \]

This gives us the solutions:

\[ x = 0 \quad \text{or} \quad x + 8 = 0 \implies x = -8 \]

Thus, the x-intercepts are

\[ \boxed{-8, 0} \]

Final Answer