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 $(0, -2)$.

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).