Question

Describe the shape of the graph of the cubic function by determining the end behavior and number of turning points. y=2x^3
What is the end behavior of the graph of the function?
1)up and up
2)down and down
3)down and up
4)up and down

How many turning points are there?(Type a whole number)

Answer 1

To analyze the cubic function \( y = 2x^3 \), we can determine both the end behavior and the number of turning points.

### End Behavior:
For a cubic function of the form \( y = ax^3 \) (where \( a \) is a positive constant), the end behavior can be described as follows:

- As \( x \to -\infty \) (left side of the graph), \( y \to -\infty \)
- As \( x \to +\infty \) (right side of the graph), \( y \to +\infty \)

Therefore, the end behavior of the graph of the function is **down and up**. This matches with option 3: **down and up**.

### Number of Turning Points:
A cubic function can have at most \( n - 1 \) turning points, where \( n \) is the degree of the polynomial. For the cubic function \( y = 2x^3 \), the degree \( n = 3 \).

Thus, the maximum number of turning points is:
\[
3 - 1 = 2
\]

However, for the specific function \( y = 2x^3 \), which is a simple monomial with no linear or constant terms added, it does not have any turning points.

So, the number of turning points is **0**.

### Summary:
- **End behavior**: Down and up (Option 3)
- **Number of turning points**: 0