The problem in the image asks to graph the function:

$y = \frac{3}{2}x^3$

It also instructs you to plot five points on the graph:

1. One point at $x = 0$.
2. Two points with negative $x$-values.
3. Two points with positive $x$-values.

Step-by-Step Solution:

The given function is a cubic function, which means its graph will resemble an "S" curve. Let's calculate the required points.

1. For $x = 0$:

$y = \frac{3}{2} \cdot 0^3 = 0$ So, the point is $(0, 0)$.

2. For negative $x$-values (e.g., $x = -2$ and $x = -1$):

$y = \frac{3}{2} \cdot (-2)^3 = \frac{3}{2} \cdot (-8) = -12$ So, the point is $(-2, -12)$.

$y = \frac{3}{2} \cdot (-1)^3 = \frac{3}{2} \cdot (-1) = -\frac{3}{2} = -1.5$ So, the point is $(-1, -1.5)$.

3. For positive $x$-values (e.g., $x = 1$ and $x = 2$):

$y = \frac{3}{2} \cdot 1^3 = \frac{3}{2} \cdot 1 = 1.5$ So, the point is $(1, 1.5)$.

$y = \frac{3}{2} \cdot 2^3 = \frac{3}{2} \cdot 8 = 12$ So, the point is $(2, 12)$.

Summary of Points:

• $(0, 0)$
• $(-2, -12)$
• $(-1, -1.5)$
• $(1, 1.5)$
• $(2, 12)$

Now you can plot these points on the graph provided.

Would you like further details or have any other questions?

Here are 5 follow-up questions you might explore:

1. How does changing the coefficient of $x^3$ affect the shape of the graph?
2. What is the general shape of a cubic function like $y = ax^3$?
3. How would the graph change if the power of $x$ were even instead of odd?
4. Can you identify the end behavior of this cubic function as $x$ approaches $\infty$ and $-\infty$?
5. What would happen to the graph if a constant term was added to the function $y = \frac{3}{2}x^3 + C$?

Tip: For cubic functions, the graph always has one inflection point where the curve changes concavity.