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Let's go through each question step by step based on the given function \( f(x) = 3x^2 - x - 2 \):

### (a) Is the point (2,8) on the graph of \( f \)?
To check if the point (2, 8) lies on the graph, we substitute \( x = 2 \) into \( f(x) \) and see if \( f(2) = 8 \).

\[
f(2) = 3(2)^2 - 2 - 2 = 3(4) - 2 - 2 = 12 - 2 - 2 = 8
\]

Since \( f(2) = 8 \), the point (2, 8) is **on the graph** of \( f \).

### (b) If \( x = -1 \), what is \( f(x) \)? What point is on the graph of \( f \)?

Substitute \( x = -1 \) into the function:

\[
f(-1) = 3(-1)^2 - (-1) - 2 = 3(1) + 1 - 2 = 3 + 1 - 2 = 2
\]

Thus, \( f(-1) = 2 \), so the point \( (-1, 2) \) is on the graph of \( f \).

### (c) If \( f(x) = -2 \), what is \( x \)? What point(s) are on the graph of \( f \)?

We solve the equation \( f(x) = -2 \):

\[
3x^2 - x - 2 = -2
\]

Simplifying:

\[
3x^2 - x = 0
\]

Factor the equation:

\[
x(3x - 1) = 0
\]

Thus, \( x = 0 \) or \( 3x - 1 = 0 \), which gives \( x = \frac{1}{3} \).

Therefore, the points are \( (0, -2) \) and \( \left( \frac{1}{3}, -2 \right) \).

### (d) What is the domain of \( f \)?

The function \( f(x) = 3x^2 - x - 2 \) is a quadratic function, and quadratic functions are defined for all real numbers. Therefore, the domain of \( f \) is **all real numbers** or \( (-\infty, \infty) \).

### (e) List the x-intercept(s), if any, of the graph of \( f \).

To find the x-intercepts, set \( f(x) = 0 \) and solve for \( x \):

\[
3x^2 - x - 2 = 0
\]

We can solve this quadratic using the quadratic formula:

\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(3)(-2)}}{2(3)}
\]

Simplifying:

\[
x = \frac{1 \pm \sqrt{1 + 24}}{6} = \frac{1 \pm \sqrt{25}}{6} = \frac{1 \pm 5}{6}
\]

Thus:

\[
x = \frac{1 + 5}{6} = 1 \quad \text{or} \quad x = \frac{1 - 5}{6} = -\frac{2}{3}
\]

The x-intercepts are \( (1, 0) \) and \( \left( -\frac{2}{3}, 0 \right) \).

### (f) List the y-intercept, if any, of the graph of \( f \).

The y-intercept occurs when \( x = 0 \). Substituting \( x = 0 \) into the function:

\[
f(0) = 3(0)^2 - 0 - 2 = -2
\]

Thus, the y-intercept is \( (0, -2) \).

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Would you like further details on any of these steps, or any other explanations?

### Related Questions:
1. How can we use the vertex form to find the vertex of this quadratic function?
2. What is the axis of symmetry for the function \( f(x) = 3x^2 - x - 2 \)?
3. How do you determine if the parabola opens upward or downward?
4. How would you find the maximum or minimum value of \( f(x) \)?
5. Can you determine the intervals where the function is increasing or decreasing?

### Tip:
When solving quadratic equations, the discriminant \( b^2 - 4ac \) can tell you if there are 0, 1, or 2 real solutions (x-intercepts).

Answer the questions about the following function.

f(x) = 3x^2 - x - 2

(a) Is the point (2,8) on the graph of f?
(b) If x = -1, what is f(x)? What point is on the graph of f?
(c) If f(x) = -2, what is x? What point(s) are on the graph of f?
(d) What is the domain of f?
(e) List the x-intercept(s), if any, of the graph of f.
(f) List the y-intercept, if any, of the graph of f.

Explore the quadratic function f(x) = 3x^2 - x - 2 through various questions such as identifying points on the graph, solving for x when f(x) = -2, and determining the domain, x-intercepts, and y-intercepts of the function. Suitable for students in Grades 9-12.

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