Let's work through the given problem, which involves analyzing the quadratic function $f(x) = x^2 - 4x - 60$. I'll address each part one by one:

a) Determine the vertex.

The vertex of a parabola given by $f(x) = ax^2 + bx + c$ can be found using the formula: $x = -\frac{b}{2a}$ Here, $a = 1$ and $b = -4$, so: $x = -\frac{-4}{2 \cdot 1} = \frac{4}{2} = 2$ To find the y-coordinate of the vertex, substitute $x = 2$ into the function: $f(2) = (2)^2 - 4(2) - 60 = 4 - 8 - 60 = -64$ So, the vertex is $(2, -64)$.

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

The graph of a quadratic function $f(x) = ax^2 + bx + c$ opens upward if $a > 0$ and downward if $a < 0$. Here, $a = 1$, which is positive, so the graph opens upward.

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

The axis of symmetry can be found from the x-coordinate of the vertex: $x = 2$ So, the equation of the axis of symmetry is $x = 2$.

d) Find any x-intercepts.

To find the x-intercepts, set $f(x) = 0$: $x^2 - 4x - 60 = 0$ This is a quadratic equation that we can solve using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Substitute $a = 1$, $b = -4$, and $c = -60$: $x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot (-60)}}{2 \cdot 1}$ $x = \frac{4 \pm \sqrt{16 + 240}}{2}$ $x = \frac{4 \pm \sqrt{256}}{2}$ $x = \frac{4 \pm 16}{2}$ So, the solutions are: $x = \frac{4 + 16}{2} = 10 \quad \text{and} \quad x = \frac{4 - 16}{2} = -6$ The x-intercepts are $(10, 0)$ and $(-6, 0)$.

e) Find the y-intercept.

The y-intercept occurs when $x = 0$: $f(0) = 0^2 - 4(0) - 60 = -60$ So, the y-intercept is $(0, -60)$.

g) State the domain and range in interval notation.

• The domain of a quadratic function is all real numbers: $\text{Domain: } (-\infty, \infty)$
• Since the parabola opens upward and has a vertex at $(2, -64)$, the range is: $\text{Range: } [-64, \infty)$

Summary of Answers:

• a) Vertex: $(2, -64)$
• b) The graph opens upward.
• c) Axis of symmetry: $x = 2$
• d) X-intercepts: $(10, 0)$ and $(-6, 0)$
• e) Y-intercept: $(0, -60)$
• g) Domain: $(-\infty, \infty)$, Range: $[-64, \infty)$

Would you like more details or have any questions on these solutions? Here are five related questions that might be helpful to explore:

1. How do you derive the formula for the vertex of a parabola?
2. Why does the sign of $a$ determine whether a parabola opens up or down?
3. What is the relationship between the vertex and the axis of symmetry?
4. How can the discriminant $b^2 - 4ac$ help determine the number of x-intercepts?
5. How do you graph a quadratic function step-by-step?

Tip: When solving quadratic equations, the quadratic formula is useful for finding exact solutions, especially when the factors aren't obvious.