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