The cubic function y = -2x^3 exhibits a downward curve, has two local extrema at (-2, 4) and (2, -4), and demonstrates distinct end behaviors as x approaches positive or negative infinity. Here option C is correct.
The function graphed below is y = -2x^3. This is the correct option among the three given choices. Here are some key points about this function and its graph:
Cubic function: This is a type of polynomial function with the highest degree of 3. It has the general form of y = ax^3 + bx^2 + cx + d, where a, b, c, d are constants and a is not zero.
Negative coefficient: The coefficient of x^3 is negative, which means the function is decreasing as x increases. This also affects the shape and direction of the graph, making it curve downwards from the top left to the bottom right.
Local extrema: The function has two local extrema, which are the highest and lowest points on the graph. They are located at (-2, 4) and (2, -4), respectively. These points can be found by solving the first derivative of the function for zero.
End behavior: The function has different behaviors as x approaches positive or negative infinity. As x goes to positive infinity, y goes to negative infinity. As x goes to negative infinity, y goes to positive infinity. This can be seen by the arrows at the ends of the graph. Here option C is correct.