Describe the shape of the graph of the cubic function by determining the end behavior and number of turning points.
y=-4x^3
The cubic function y = -4x^3 has an odd degree (3) and a negative leading coefficient (-4).
This means that as x approaches negative infinity, the function value also approaches negative infinity, and as x approaches positive infinity, the function value also approaches positive infinity. The end behavior can be described as "downwards on the left and upwards on the right."
The number of turning points of a cubic function is determined by the degree of the function minus 1. In this case, the cubic function has 3-1=2 turning points.
To describe the shape of the graph of the cubic function y = -4x^3, we need to determine the end behavior and the number of turning points.
1. End Behavior:
The end behavior of a function describes how the graph behaves as x approaches positive or negative infinity. In this case, the leading term of the equation is -4x^3, which means that as x approaches infinity (positive or negative), the value of y will also approach infinity (positive or negative). Therefore, the end behavior of the graph is that it goes up to positive infinity on one side and down to negative infinity on the other side.
2. Number of Turning Points:
The turning points of a graph occur where the graph changes from increasing to decreasing or vice versa. For a cubic function like y = -4x^3, there will always be two turning points.
In this particular case, since the coefficient of x^3 is negative (-4), the graph will first rise from negative infinity, reach a maximum turning point, and then descend to negative infinity. It will then rise again, reach a minimum turning point, and finally descend to positive infinity.
To summarize:
- The end behavior of the graph is that it goes up to positive infinity on one side and down to negative infinity on the other side.
- There are two turning points: one maximum and one minimum.
Please note that this description assumes a general cubic function. Actual graphs may be subject to horizontal and vertical shifts, and stretches or compressions based on additional terms or coefficient values.
To describe the shape of the graph of the cubic function y = -4x^3, we can analyze its end behavior and determine the number of turning points.
End Behavior:
1. For large positive values of x: When x approaches positive infinity, the value of -4x^3 becomes increasingly negative. Therefore, as x goes towards positive infinity, the y-values decrease without bound, meaning the graph heads down towards negative infinity in the upper-right quadrant.
2. For large negative values of x: Similarly, when x approaches negative infinity, -4x^3 becomes increasingly negative. Thus, as x goes towards negative infinity, the y-values also decrease without bound, indicating that the graph extends downwards towards negative infinity in the lower-left quadrant.
Number of Turning Points:
To determine the number of turning points, we need to consider the degree of the polynomial. The given cubic function has a degree of 3, meaning it is a cubic polynomial. Cubic polynomials can have a maximum of two turning points.
Since the function y = -4x^3 is a cubic polynomial, it is guaranteed to have either one or two turning points. To find out the exact number, we need to examine the graph or use more advanced mathematical techniques. One approach is to take the derivative of the function y = -4x^3 and analyze where it equals zero to determine critical points. However, in this case, since the cubic function has a simple form, we can observe that it will have only one turning point, which is a local maximum.
Therefore, the graph of the function y = -4x^3 is a cubic curve that declines in both upper-right and lower-left quadrants, and it has a single turning point, specifically a local maximum.