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Let's go through each part of the problem based on the function \( c(x) = 3x^2 - 6x - 1 \). Here is how to address each question:

### Part (a) Write the function in vertex form:
The function is already shown in the vertex form in your answer:
\[
c(x) = 3(x - 1)^2 - 4
\]
The vertex form of a quadratic function is typically written as \( c(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex.

### Part (b) Identify the vertex:
From the vertex form \( c(x) = 3(x - 1)^2 - 4 \), the vertex is \( (h, k) = (1, -4) \).

### Part (c) Determine the x-intercept(s):
To find the x-intercepts, we set \( c(x) = 0 \) and solve for \( x \):
\[
3(x - 1)^2 - 4 = 0
\]
First, isolate the squared term:
\[
3(x - 1)^2 = 4
\]
\[
(x - 1)^2 = \frac{4}{3}
\]
Take the square root of both sides:
\[
x - 1 = \pm \sqrt{\frac{4}{3}} = \pm \frac{2\sqrt{3}}{3}
\]
Thus, the x-intercepts are:
\[
x = 1 \pm \frac{2\sqrt{3}}{3}
\]

### Part (d) Determine the y-intercept(s):
To find the y-intercept, set \( x = 0 \):
\[
c(0) = 3(0)^2 - 6(0) - 1 = -1
\]
So, the y-intercept is \( (0, -1) \).

### Part (e) Sketch the function:
The function is a parabola opening upwards (since the coefficient of \( x^2 \) is positive). It has a vertex at \( (1, -4) \), and it intersects the y-axis at \( (0, -1) \).

### Part (f) Determine the axis of symmetry:
The axis of symmetry is the vertical line that passes through the vertex. Therefore, the axis of symmetry is:
\[
x = 1
\]

### Part (g) Determine the minimum or maximum value:
Since the parabola opens upwards, the vertex represents the minimum point. The minimum value is \( -4 \), which occurs at \( x = 1 \).

### Part (h) Write the domain and range in interval notation:
- The domain of any quadratic function is \( (-\infty, \infty) \).
- The range starts at the minimum value, \( -4 \), and goes to infinity, so the range is \( [-4, \infty) \).

### Summary:
1. **Vertex form:** \( 3(x - 1)^2 - 4 \)
2. **Vertex:** \( (1, -4) \)
3. **X-intercepts:** \( x = 1 \pm \frac{2\sqrt{3}}{3} \)
4. **Y-intercept:** \( (0, -1) \)
5. **Axis of symmetry:** \( x = 1 \)
6. **Minimum value:** \( -4 \) at \( x = 1 \)
7. **Domain:** \( (-\infty, \infty) \)
8. **Range:** \( [-4, \infty) \)

Do you need more details or explanations on any step?

---

### Five follow-up questions:
1. How do you complete the square to convert a quadratic equation to vertex form?
2. What is the significance of the axis of symmetry in a quadratic function?
3. How do you find the discriminant, and what does it tell you about the x-intercepts?
4. Can you explain why the function has a minimum rather than a maximum?
5. How does the value of the coefficient of \( x^2 \) affect the shape of the parabola?

### Tip:
Remember that the vertex form makes it easier to identify the vertex and the direction of the parabola's opening!

Consider the function c(x) = 3x^2 - 6x - 1. Write the function in vertex form, identify the vertex, determine intercepts, sketch the function, and analyze key properties.

Learn how to convert the quadratic function c(x) = 3x^2 - 6x - 1 into vertex form, find the intercepts, vertex, axis of symmetry, and analyze the function's domain and range. Suitable for Grades 9-11, this problem covers key algebraic concepts.

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