Describe the shape of the graph of the cubic function by determining the end behavior and number of turning points.
y=4x^3-3
The given cubic function is y = 4x^3 - 3.
End behavior:
The leading term in the function is 4x^3, which means that as x approaches positive infinity, the function also approaches positive infinity. And as x approaches negative infinity, the function approaches negative infinity. This indicates that the graph of the cubic function will have opposite end behavior on either side of the y-axis.
Number of turning points:
Since the degree of the polynomial is 3, we expect there to be 3 turning points on the graph of the cubic function.
To describe the shape of the graph of the cubic function y = 4x^3 - 3, we can determine its end behavior and the number of turning points.
1. End Behavior:
The end behavior refers to the direction in which the graph of the function approaches as x approaches positive infinity (∞) and negative infinity (-∞).
- Since the leading term of the function is positive (4x^3), as x approaches -∞, the graph will be going down towards negative infinity.
- Similarly, as x approaches +∞, the graph will be going up towards positive infinity.
2. Number of Turning Points:
The number of turning points on a cubic function can be determined by the degree or power of the polynomial, which is 3 in this case. A cubic function can have zero, one, or two turning points.
To find the number of turning points, we need to calculate the derivative of the function and determine the number of solutions for the equation f'(x) = 0.
Taking the derivative of y = 4x^3 - 3, we get:
f'(x) = 12x^2
Setting f'(x) = 0 and solving for x, we have:
12x^2 = 0
x^2 = 0
x = 0 (double root)
Since the derivative has one double root (x = 0), there is one turning point on the graph of the cubic function.
In summary, the graph of the cubic function y = 4x^3 - 3 will have one turning point and will go down towards negative infinity as x approaches -∞ and go up towards positive infinity as x approaches +∞.
To describe the shape of the graph of the cubic function y = 4x^3 - 3, we need to determine the end behavior and the number of turning points.
1. End Behavior:
The end behavior of a polynomial function is determined by the highest power term and its coefficient. In this case, the highest power term is 4x^3.
Since the coefficient of the highest power term is positive (4 > 0), the end behavior of the graph is as follows:
- When x approaches negative infinity, y approaches negative infinity (∞).
- When x approaches positive infinity, y approaches positive infinity (∞).
2. Turning Points:
The number of turning points in a cubic function can be determined by the degree of the function, which is 3. A cubic function can have up to two turning points.
To find the turning points, we need to find the x-coordinates where the derivative of the function equals zero. Let's find the derivative of y = 4x^3 - 3:
dy/dx = 12x^2
Now, let's solve for dy/dx = 0:
12x^2 = 0
x^2 = 0
From this equation, we can see that x = 0 is the only solution. Therefore, there is only one turning point at x = 0.
In summary:
- The end behavior of the graph is as follows: When x approaches negative infinity, y approaches negative infinity (∞), and when x approaches positive infinity, y approaches positive infinity (∞).
- The graph has one turning point at x = 0.