Analyzing and Sketching the Graph of y x^3 - 2x^2

Understanding the behavior and sketching the graph of the function y x^3 - 2x^2 involves identifying its key features such as intercepts, turning points, and the overall shape. This analysis will help in creating an accurate and comprehensive sketch of the graph.

Determining Key Features

Let's begin by finding the derivative of the function y x^3 - 2x^2. The derivative, denoted as dy/dx, is calculated as follows:

dy/dx 3x^2 - 4x

The derivative provides important information about the function's behavior:

The function is increasing where dy/dx > 0, which occurs in intervals (-∞, 0) U (4/3, ∞). The function is decreasing where dy/dx , which occurs in the interval (0, 4/3).

Identifying Critical Points

To find the critical points, we need to set the derivative to zero and solve for x:

3x^2 - 4x 0

This can be factored as:

x(3x - 4) 0

Thus, the critical points are x 0 and x 4/3.

Determining Turning Points and Extrema

These critical points can be further classified by examining the second derivative:

d^2y/dx^2 6x - 4

Evaluating the second derivative at the critical points:

d^2y/dx^2 at x 0: 6(0) - 4 -4 (concave down, indicating a maximum)

d^2y/dx^2 at x 4/3: 6(4/3) - 4 8 - 4 4 (concave up, indicating a minimum)

Given these results, we can conclude that there is a local maximum at x 0 and a local minimum at x 4/3.

Intercepts and Sketching the Graph

The x-intercepts of the function are the values of x for which y 0:

x^3 - 2x^2 0

This can be factored as:

x^2(x - 2) 0

Thus, the x-intercepts are x 0 and x 2.

Similarly, the y-intercept is the value of y when x 0, which is y 0.

Now, let's sketch the graph based on the identified features:

Local maximum at (0, 0) Local minimum at (4/3, -16/27) x-intercepts at (0, 0) and (2, 0) Behavior for large x: The x^3 term dominates, causing the graph to rise sharply for positive and negative values of x.

Using these key points and the overall shape of the cubic function, we can sketch the graph accurately. This approach ensures that the curve is correctly depicted with its critical points and behavior at infinity.

Conclusion

In summary, understanding the key features of the function y x^3 - 2x^2 through its derivative and critical points allows for the accurate sketching of its graph. Identifying the intercepts, critical points, and the behavior of the function in various intervals provides a clear picture of its shape and key characteristics.