Let's analyze and solve the problem based on the quadratic function provided:

$f(x) = x^2 - 4x - 12$

a) Determine the vertex:

The formula to find the vertex of a parabola $f(x) = ax^2 + bx + c$ is given by: $x = -\frac{b}{2a}$ Here, $a = 1$, $b = -4$, and $c = -12$. Let's calculate the x-coordinate of the vertex: $x = -\frac{-4}{2(1)} = \frac{4}{2} = 2$ To find the y-coordinate, substitute $x = 2$ back into the equation for $f(x)$: $f(2) = (2)^2 - 4(2) - 12 = 4 - 8 - 12 = -16$ So, the vertex is $(2, -16)$.

b) Does the graph "open up" or "open down"?

Since the coefficient of $x^2$ (i.e., $a = 1$) is positive, the parabola opens upward.

c) What is the equation of the axis of symmetry?

The axis of symmetry is a vertical line that passes through the vertex. The equation is: $x = 2$

d) Find any x-intercepts:

To find the x-intercepts, set $f(x) = 0$ and solve for $x$: $x^2 - 4x - 12 = 0$ Using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Substitute $a = 1$, $b = -4$, and $c = -12$: $x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-12)}}{2(1)} = \frac{4 \pm \sqrt{16 + 48}}{2} = \frac{4 \pm \sqrt{64}}{2} = \frac{4 \pm 8}{2}$ So, the solutions are: $x = \frac{4 + 8}{2} = 6 \quad \text{and} \quad x = \frac{4 - 8}{2} = -2$ Thus, the x-intercepts are $x = 6$ and $x = -2$.

e) Find the y-intercept:

To find the y-intercept, substitute $x = 0$ into the equation: $f(0) = 0^2 - 4(0) - 12 = -12$ So, the y-intercept is $(0, -12)$.

f) Sketch the graph:

The graph is a parabola opening upwards with the following key points:

• Vertex at $(2, -16)$
• X-intercepts at $(-2, 0)$ and $(6, 0)$
• Y-intercept at $(0, -12)$

g) State the domain and range in interval notation:

• The domain of any quadratic function is all real numbers: $(-\infty, \infty)$.
• The range is all values of $y$ greater than or equal to the minimum value (the y-coordinate of the vertex): $[-16, \infty)$.

Do you want further clarification on any of these steps or a deeper explanation of any part?

Here are 5 related questions you might want to explore:

1. How do you use the discriminant to determine the number of x-intercepts for a quadratic function?
2. Can you explain the relationship between the vertex and the axis of symmetry?
3. How does changing the value of $a$ affect the graph of a parabola?
4. What is the significance of the y-intercept in graphing quadratic functions?
5. How do you derive the quadratic formula?

Tip: Always check the sign of the leading coefficient $a$ to determine whether a parabola opens upward (positive) or downward (negative).