Let's analyze the quadratic function provided:

$f(x) = x^2 - 8x - 9$

1. Axis of Symmetry:

The axis of symmetry of a quadratic function $f(x) = ax^2 + bx + c$ is given by the formula:

$x = -\frac{b}{2a}$

For $f(x) = x^2 - 8x - 9$, we have $a = 1$ and $b = -8$. Plugging these values into the formula:

$x = -\frac{-8}{2(1)} = \frac{8}{2} = 4$

Thus, the axis of symmetry is $x = 4$.

2. Vertex:

The vertex is the point on the graph where the parabola turns. The x-coordinate of the vertex is the axis of symmetry, $x = 4$.

Now, substitute $x = 4$ into the equation of the function to find the y-coordinate of the vertex:

$f(4) = (4)^2 - 8(4) - 9 = 16 - 32 - 9 = -25$

Thus, the vertex is at $(4, -25)$.

3. Domain:

The domain of any quadratic function is always all real numbers, because the function is defined for all values of $x$. In interval notation:

$\text{Domain of } f: (-\infty, \infty)$

4. Range:

Since the parabola opens upwards (the coefficient of $x^2$ is positive), the range starts at the y-coordinate of the vertex and goes to infinity. The vertex's y-coordinate is $-25$, so the range is:

$\text{Range of } f: [-25, \infty)$

Summary:

• Axis of symmetry: $x = 4$
• Domain: $(-\infty, \infty)$
• Range: $[-25, \infty)$

Would you like help graphing this equation or exploring any further details? Here are five relative questions that can expand this information:

1. What is the process to complete the square for this function?
2. How do you determine the intercepts for this quadratic function?
3. Can you describe how transformations (translations or reflections) affect the graph of $f(x)$?
4. What does the discriminant of a quadratic function tell you about its roots?
5. How would you find the maximum or minimum value of a more general quadratic function?

Tip: When solving quadratics, always start by identifying the vertex using symmetry—it provides key insights into the function's graph and behavior.