Determining The Slant Asymptote Of F(x) = (8x³ - 6x² + 4) / (2x² + 2x - 4)

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To determine the slant asymptote of the rational function f(x) = (8x³ - 6x² + 4) / (2x² + 2x - 4), we need to perform polynomial long division. Slant asymptotes, also known as oblique asymptotes, occur when the degree of the numerator is exactly one greater than the degree of the denominator. In this case, the numerator has a degree of 3 (8x³) and the denominator has a degree of 2 (2x²), so a slant asymptote exists. Identifying this slant asymptote is crucial for understanding the behavior of the function as x approaches positive or negative infinity. The slant asymptote provides a linear approximation of the function's end behavior, making it an invaluable tool in graphing and analyzing rational functions. To find it, we'll divide the numerator by the denominator, focusing on the quotient, which will give us the equation of the asymptote. The remainder is not considered when determining the slant asymptote.

Polynomial Long Division

Let's perform polynomial long division to divide 8x³ - 6x² + 4 by 2x² + 2x - 4. This process involves dividing the leading term of the dividend (8x³) by the leading term of the divisor (2x²) to obtain the first term of the quotient. Then, we multiply the entire divisor by this term and subtract the result from the dividend. We repeat this process with the new polynomial until the degree of the remainder is less than the degree of the divisor. In this specific case, dividing 8x³ by 2x² gives us 4x, which will be the first term of our quotient. This long division method systematically breaks down the complex rational function into simpler components, revealing the linear asymptote that governs the function's behavior at extreme values. Polynomial long division isn't just a mechanical procedure; it's a powerful algebraic technique for unraveling the structure of rational expressions.

Step-by-step Long Division:

  1. Divide 8x³ by 2x² to get 4x.
  2. Multiply (2x² + 2x - 4) by 4x: 4x * (2x² + 2x - 4) = 8x³ + 8x² - 16x.
  3. Subtract (8x³ + 8x² - 16x) from (8x³ - 6x² + 4): (8x³ - 6x² + 4) - (8x³ + 8x² - 16x) = -14x² + 16x + 4.
  4. Divide -14x² by 2x² to get -7.
  5. Multiply (2x² + 2x - 4) by -7: -7 * (2x² + 2x - 4) = -14x² - 14x + 28.
  6. Subtract (-14x² - 14x + 28) from (-14x² + 16x + 4): (-14x² + 16x + 4) - (-14x² - 14x + 28) = 30x - 24.

The quotient is 4x - 7, and the remainder is 30x - 24. Therefore, the result of the division can be written as:

f(x) = 4x - 7 + (30x - 24) / (2x² + 2x - 4)

Determine the Slant Asymptote

From the long division, we found that the rational function can be expressed as f(x) = 4x - 7 + (30x - 24) / (2x² + 2x - 4). The slant asymptote is determined by the quotient obtained from the long division, which is 4x - 7. As x approaches positive or negative infinity, the term (30x - 24) / (2x² + 2x - 4) approaches zero because the degree of the denominator (2) is greater than the degree of the numerator (1). The slant asymptote, therefore, is the line y = 4x - 7. This linear equation describes the function's behavior at extreme values, providing a valuable tool for sketching the graph and analyzing its end behavior. The slant asymptote is not a point or a region; it is a line that the function approaches but never quite reaches. Understanding how to determine this asymptote is essential for a complete analysis of rational functions.

Thus, the slant asymptote of the given rational function f(x) = (8x³ - 6x² + 4) / (2x² + 2x - 4) is y = 4x - 7. This means that as x gets very large (positive or negative), the graph of f(x) will get closer and closer to the line y = 4x - 7. This asymptotic behavior is a fundamental characteristic of rational functions, especially those where the degree of the numerator exceeds the degree of the denominator by one. The ability to identify and interpret these asymptotes is crucial for understanding the overall behavior of the function and its graph. By focusing on the quotient obtained from polynomial long division, we can effectively determine the equation of the slant asymptote and gain valuable insights into the function's long-term trends.

Verification and Graphing

To verify our result, we can graph the function f(x) = (8x³ - 6x² + 4) / (2x² + 2x - 4) and the line y = 4x - 7. As x approaches positive or negative infinity, the graph of f(x) should get closer and closer to the line y = 4x - 7. Graphing is a powerful tool for confirming our algebraic calculations and visualizing the asymptotic behavior of the function. A visual representation can often provide a deeper understanding of the mathematical concepts involved. In this case, the graph will clearly show how the slant asymptote acts as a guide for the function's end behavior. This verification step is crucial in ensuring the accuracy of our solution and reinforcing the connection between algebraic and graphical representations of functions.

Graphing Tools:

Various online graphing calculators and software can be used to plot the function and its asymptote. Some popular tools include Desmos, GeoGebra, and Wolfram Alpha. These tools allow for precise graphing and can help visualize the relationship between the function and its slant asymptote. By using these tools, we can not only verify our calculations but also gain a better understanding of the function's overall behavior, including its intercepts, local maxima and minima, and other key features. The use of graphing technology is an integral part of modern mathematics education and research, providing a visual means to explore and confirm algebraic results.

Interpreting the Graph:

When graphing, observe how the function approaches the line y = 4x - 7 as x becomes very large or very small. The graph should oscillate around the line, getting closer and closer but never actually touching it. This behavior is characteristic of slant asymptotes. Additionally, note any vertical asymptotes, which occur where the denominator of the rational function is equal to zero. These vertical asymptotes, along with the slant asymptote, provide a comprehensive picture of the function's asymptotic behavior. The graph serves as a visual confirmation of our algebraic analysis, reinforcing the concept of asymptotic behavior and the role of the slant asymptote in determining the function's end behavior.

Conclusion

In conclusion, the slant asymptote of the rational function f(x) = (8x³ - 6x² + 4) / (2x² + 2x - 4) is y = 4x - 7. This was determined by performing polynomial long division and identifying the quotient, which represents the equation of the slant asymptote. Understanding how to find slant asymptotes is essential for analyzing the behavior of rational functions, particularly as x approaches infinity. The slant asymptote provides a linear approximation of the function's end behavior, making it a valuable tool for graphing and problem-solving. This process not only reinforces algebraic skills but also deepens the understanding of how functions behave at extreme values. The combination of algebraic manipulation, such as polynomial long division, and graphical verification provides a comprehensive approach to understanding rational functions and their asymptotes. Mastering these techniques is crucial for success in calculus and other advanced mathematical fields. Ultimately, the ability to determine the slant asymptote allows for a more complete and nuanced understanding of the rational function's characteristics and its behavior across the entire domain.