A Comprehensive Guide to Sketching the Graph of yx^3-2x^2

The function ( y x^3 - 2x^2 ) is a cubic polynomial. Understanding how to sketch its graph is crucial in algebra and calculus. This guide provides a step-by-step process to graph this function effectively, both manually and using modern software tools.

Understanding the Function

The given function is ( y x^3 - 2x^2 ). This function is a cubic polynomial, meaning it has a degree of three. Let's break down the function and understand its key features.

Y-Intercept

To find the y-intercept, set ( x 0 ) in the function:

[ y 0^3 - 2 cdot 0^2 0 ]

Hence, the y-intercept is at the point (0, 0).

X-Intercepts

To find the x-intercepts, set ( y 0 ) and solve for ( x ):

[ x^3 - 2x^2 0 ]

This can be factored as:

[ x^2(x - 2) 0 ]

So, the x-intercepts are at ( x 0 ) (double root) and ( x 2 ).

Key Points for Graphing

Key points in graphing this function include the y-intercept, x-intercepts, and local extrema. Let's explore these in detail.

Local Extremum

To find the local extrema, we first need the first derivative of the function:

[ y' frac{d}{dx}(x^3 - 2x^2) 3x^2 - 4x ]

Set the first derivative to zero to find the critical points:

[ 3x^2 - 4x 0 ]

This can be factored as:

[ x(3x - 4) 0 ]

So, the critical points are ( x 0 ) and ( x frac{4}{3} ).

Second Derivative Test

Next, we use the second derivative to determine the nature of these critical points:

[ y'' frac{d}{dx}(3x^2 - 4x) 6x - 4 ]

Evaluate the second derivative at the critical points:

- For ( x 0 ): [ y''(0) 6(0) - 4 -4 ] (local maximum) - For ( x frac{4}{3} ): [ y''left(frac{4}{3}right) 6left(frac{4}{3}right) - 4 4 ] (local minimum)

Thus, there is a local maximum at ( (0, 0) ) and a local minimum at ( left( frac{4}{3}, -frac{32}{27} right) ).

Sketching the Graph

To sketch the graph, you can start by plotting the key points and using the continuity of polynomial functions. The graph will pass through the x-intercepts and will have a turning point on the x-axis at ( x 0 ).

Graph Sketching Steps

1. Plot the y-intercept at (0, 0). 2. Plot the x-intercepts at (0, 0) (double root) and (2, 0). 3. Identify the local maximum at (0, 0). 4. Identify the local minimum at ( left( frac{4}{3}, -frac{32}{27} right) ). 5. Draw the curve such that it passes through these points and follows the behavior of a cubic function (one change of direction at a local extremum).

Using Software for Graphing

For a more accurate and time-saving approach, you can use software like R to graph the function. Here’s a brief guide on how to do it in R:

R Code Example

Here is a sample R code to plot the graph of ( y x^3 - 2x^2 ):

```r x This code generates a smooth plot of the function and marks the key points of interest.

Conclusion

Sketching the graph of ( y x^3 - 2x^2 ) involves understanding the function's intercepts and extrema. Using software tools like R can significantly enhance the accuracy and efficiency of the graphing process. Whether using pen and paper or digital tools, the key is to focus on the essential features of the function.