Graphing the Function y x^4 - 4x^2 - 4: A Comprehensive Guide for SEO

Have you ever found yourself pondering the graph of the function y x^4 - 4x^2 - 4? This polynomial function might seem daunting at first, but fear not! This article will guide you through the process of sketching its graph with respect to the x-axis. Let's dive into the details.

Understanding the Function

Given the function y x^4 - 4x^2 - 4, we can determine its graph's key features by following a systematic approach. Start by setting the function equal to zero:

Setting the Equation to Zero

0 x^4 - 4x^2 - 4

From this step, we can identify the x-intercepts and determine how the curve behaves around these points. Let's break it down further:

1. x-intercepts

x 0 x -2

Here, we see that the curve crosses the x-axis at x 0 and touches the x-axis at x -2. The reason for this behavior is due to the multiplicity of the roots. Specifically, x 0 is a root of multiplicity 2, meaning the curve touches the x-axis at this point. In contrast, x -2 is a single root, so the curve crosses the x-axis at this point.

For a polynomial function, the end behaviors can help us predict the general shape of the graph:

2. End Behaviors

The highest-degree term of the polynomial is x^4, which has a positive coefficient of 1. This means:

The curve will tend towards negative infinity along the y-axis as x approaches negative infinity. The curve will tend towards positive infinity along the y-axis as x approaches positive infinity.

Step-by-Step Graphing

Now that we have a good understanding of the function's behavior, let's proceed with graphing it. Here are the steps:

1. Determine Roots and Critical Points

First, we can calculate the critical points by finding the first derivative:

y′ddx(x4?4x2?4)4x3?8x.

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

04x3?8x

Solving the equation, we find:

x 0 x -2/3 x 2/3 (Note that x 0 has multiplicity 2 as a repeated root)

2. Calculate the Function Values at Critical Points

Next, we can calculate the function values at the critical points:

y(0) -4 y(-2/3) ≈ -3.222... y(2/3) ≈ -3.222...

This helps us understand the local maximum and minimum points. Since the second derivative is y#x2032;#x2032;12x?8, we find:

At x -2/3, y'' -20/3, which is negative, indicating a local maximum. At x 2/3, y'' -20/3, which is negative, indicating a local maximum.

3. Sketch the Graph

With the roots, critical points, and end behaviors in mind, we can sketch the graph. Here’s a general path:

The curve starts from the upper left, crosses the x-axis at x 0, and touches the x-axis at x -2. The curve has a local maximum at x -2/3 and a local minimum at x 2/3. The curve tends towards positive infinity as x approaches both positive and negative infinity.

For a more accurate graph, you can use online graphing tools like Desmos or WolframAlpha.

Additional Points for Reference

To further refine the graph, calculate some additional points in the range -3 ≤ x ≤ 1:

At x -3, y 47 At x -2, y 0 At x -1, y -7 At x 0, y -4 At x 1, y -7 At x 2, y 0

These points will help you sketch the curve more accurately.

Conclusion

Graphing polynomial functions can be a powerful tool in understanding their behavior. By following the steps outlined in this guide, you can effectively graph the function y x^4 - 4x^2 - 4. For more information on polynomial functions and their graphs, refer to your textbook and practice with different functions and problems. Remember, the key to mastering graphing is practice and a deep understanding of the underlying concepts.