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The problem presented involves analyzing the quadratic function \( g(x) = 3x^2 - 18x + 26 \). Let's tackle each part step-by-step:

### (a) Write the function in vertex form.
The vertex form of a quadratic function is \( g(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex of the parabola. We can complete the square to convert the function into this form.

Start with:
\[
g(x) = 3x^2 - 18x + 26
\]
Factor out 3 from the quadratic and linear terms:
\[
g(x) = 3(x^2 - 6x) + 26
\]
Now complete the square inside the parentheses. Take half of -6 (which is -3), square it to get 9, and add and subtract 9 inside the parentheses:
\[
g(x) = 3(x^2 - 6x + 9 - 9) + 26
\]
\[
g(x) = 3((x - 3)^2 - 9) + 26
\]
Distribute the 3:
\[
g(x) = 3(x - 3)^2 - 27 + 26
\]
Simplify:
\[
g(x) = 3(x - 3)^2 - 1
\]

Thus, the function in vertex form is:
\[
g(x) = 3(x - 3)^2 - 1
\]

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

### (c) Determine the x-intercept(s).
To find the x-intercepts, set \( g(x) = 0 \) and solve for \( x \):
\[
0 = 3(x - 3)^2 - 1
\]
Add 1 to both sides:
\[
1 = 3(x - 3)^2
\]
Divide by 3:
\[
\frac{1}{3} = (x - 3)^2
\]
Take the square root of both sides:
\[
x - 3 = \pm \sqrt{\frac{1}{3}}
\]
\[
x = 3 \pm \frac{\sqrt{3}}{3}
\]

Thus, the x-intercepts are \( x = 3 + \frac{\sqrt{3}}{3} \) and \( x = 3 - \frac{\sqrt{3}}{3} \).

### (d) Determine the y-intercept.
To find the y-intercept, set \( x = 0 \) in the original equation:
\[
g(0) = 3(0)^2 - 18(0) + 26 = 26
\]

Thus, the y-intercept is \( (0, 26) \).

### (e) Sketch the function.
The graph is a parabola opening upwards (since the coefficient of \( x^2 \) is positive) with vertex at \( (3, -1) \), y-intercept at \( (0, 26) \), and x-intercepts around \( 3 \pm \frac{\sqrt{3}}{3} \). The function has axis of symmetry \( x = 3 \).

### (f) Determine the axis of symmetry.
The axis of symmetry is the vertical line through the vertex, so it is:
\[
x = 3
\]

### (g) Determine the minimum or maximum value of the function.
Since the parabola opens upwards, the function has a **minimum value** at the vertex. The minimum value is the y-coordinate of the vertex:
\[
\text{Minimum value} = -1
\]

### (h) Write the domain and range in interval notation.
- **Domain**: Since this is a quadratic function, the domain is all real numbers:
\[
\text{Domain} = (-\infty, \infty)
\]
- **Range**: The minimum value is \( -1 \), and the parabola opens upwards, so the range is:
\[
\text{Range} = [-1, \infty)
\]

---

Do you want further details or have any questions on these steps?

Here are 5 related questions to explore further:
1. How do you complete the square for a quadratic function in general?
2. What does the discriminant tell you about the number of x-intercepts for a quadratic function?
3. How do you find the vertex using the formula \( x = \frac{-b}{2a} \) from the standard form of a quadratic function?
4. How do transformations affect the graph of a quadratic function?
5. How do you determine the concavity of a quadratic function?

**Tip**: The vertex form is particularly useful for identifying the vertex, axis of symmetry, and minimum/maximum values of a quadratic function.

Consider the function g(x) = 3x^2 - 18x + 26. Write the function in vertex form. Identify the vertex. Determine the x-intercept(s), y-intercept(s). Sketch the function. Determine the axis of symmetry. Determine the minimum or maximum value of the function. Write the domain and range in interval notation.

Learn how to write the quadratic function g(x) = 3x^2 - 18x + 26 in vertex form. This problem covers identifying the vertex, finding x and y intercepts, determining the axis of symmetry, and the domain and range of the function. Detailed step-by-step explanation.

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