We are given the function:

$f(x) = 3x^3 + 3x^2 - 18x$

1. Finding the y-intercept:

The y-intercept occurs when $x = 0$. So, let's calculate $f(0)$:

$f(0) = 3(0)^3 + 3(0)^2 - 18(0) = 0$

Thus, the y-intercept is at the point $(0, 0)$.

2. Finding the x-intercepts:

The x-intercepts occur where $f(x) = 0$. We need to solve the equation:

$3x^3 + 3x^2 - 18x = 0$

Factor out the common term, which is $3x$:

$3x(x^2 + x - 6) = 0$

Now, solve $x^2 + x - 6 = 0$. This is a quadratic equation, which we can factor:

$x^2 + x - 6 = (x - 2)(x + 3)$

Thus, the factored form of the equation is:

$3x(x - 2)(x + 3) = 0$

Now, solve for $x$:

$x = 0, \quad x = 2, \quad x = -3$

So, the x-intercepts are at the points $(0, 0)$, $(2, 0)$, and $(-3, 0)$.

Summary:

• The y-intercept is at $(0, 0)$.
• The x-intercepts are at $(0, 0)$, $(2, 0)$, and $(-3, 0)$.

Would you like further details on any part of this process?

Here are 5 related questions to explore further:

1. How do we determine if a graph crosses or touches the x-axis at the intercepts?
2. How does factoring help in solving cubic equations?
3. Can we use the quadratic formula instead of factoring in this case?
4. How does the behavior of the graph change near each intercept?
5. What are the steps to check for symmetry in the graph of this function?

Tip: When solving cubic functions, always check for common factors to simplify the equation first.