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.