Describe the shape of the graph of the cubic function by determining the end behavior and number of turning points.
y=2x^3-4x-3
The end behavior of the cubic function y = 2x^3 - 4x - 3 can be determined by looking at the coefficient of the highest power term, which is 2x^3. Since the coefficient is positive, the graph will rise to positive infinity on the right side and fall to negative infinity on the left side.
To determine the number of turning points, we can find the derivative of the function and count the number of places where the derivative changes sign. The derivative of y = 2x^3 - 4x - 3 is y' = 6x^2 - 4.
Setting the derivative equal to zero, we get 6x^2 - 4 = 0. Solving this equation, we find x^2 = 2/3, and therefore, x = ±√(2/3).
Since the discriminant is positive, there are two distinct real roots. We then have two turning points: one where the graph changes from decreasing to increasing at x = -√(2/3), and another where the graph changes from increasing to decreasing at x = √(2/3).
Therefore, the graph of the cubic function y = 2x^3 - 4x - 3 has two turning points.
To describe the shape of the graph of the cubic function y = 2x^3 - 4x - 3, we need to determine its end behavior and the number of turning points.
1. End behavior:
The end behavior of a function refers to how the function behaves as x approaches positive infinity and negative infinity. To determine this, we consider the leading term of the function, which in this case is 2x^3.
Since the leading term is an odd-degree polynomial, the end behavior of the function will be in opposite directions as x approaches positive infinity and negative infinity. In other words, the graph will point upward as x becomes very large (positive infinity) and downward as x becomes very large (negative infinity).
2. Turning points:
Turning points are the points on the graph where the function changes from increasing to decreasing or vice versa. To find the turning points of the cubic function, we need to find the x-values where the derivative of the function equals zero.
First, let's find the derivative of the function:
y' = 6x^2 - 4
Next, set the derivative equal to zero and solve for x:
6x^2 - 4 = 0
6x^2 = 4
x^2 = 4/6
x^2 = 2/3
x = ±√(2/3)
So, the turning points of the cubic function occur at x = √(2/3) and x = -√(2/3).
In summary, the shape of the graph of the cubic function y = 2x^3 - 4x - 3 is as follows:
- The end behavior is upwards as x approaches positive infinity and downwards as x approaches negative infinity.
- The function has two turning points at x = √(2/3) and x = -√(2/3).
To describe the shape of the graph of a cubic function, we need to determine its end behavior and the number of turning points.
1. End behavior: The end behavior of a function describes what happens to the y-values as x approaches positive or negative infinity.
For a cubic function, the general form is f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants.
Looking at the given cubic function y = 2x^3 - 4x - 3, we can determine its end behavior by observing the coefficient of the leading term (x^3). In this case, the coefficient is 2.
If the leading coefficient is positive (as it is in this case), the end behavior of the graph will be:
- As x approaches positive infinity, y will increase without bound (goes to positive infinity).
- As x approaches negative infinity, y will decrease without bound (goes to negative infinity).
So, the graph of the cubic function y = 2x^3 - 4x - 3 will go up toward positive infinity on the right side and down toward negative infinity on the left side.
2. Turning points: The turning points of a function are the points where the graph changes direction. They can be local maximums or local minimums.
To find the number of turning points, we need to take the derivative of the given function and determine its critical points. The derivative of y = 2x^3 - 4x - 3 is:
dy/dx = 6x^2 - 4.
Setting the derivative equal to zero and solving for x, we have:
6x^2 - 4 = 0
6x^2 = 4
x^2 = 4/6
x^2 = 2/3
x = ±√(2/3).
So, the critical points occur at x = √(2/3) and x = -√(2/3).
Since the second derivative test (evaluating the concavity at these critical points) is not necessary for this question, we can conclude that there are two turning points in the graph of the cubic function y = 2x^3 - 4x - 3.
In summary, the graph of the given cubic function has an end behavior that goes up toward positive infinity on the right side and down toward negative infinity on the left side. It also has two turning points.