Question

Complete the following steps for the given functions. a. Use polynomial long division to find the slant asymptote of \(f\) b. Find the vertical asymptotes of \(f\) c. Graph \(f\) and all its asymptotes with a graphing utility. Then sketch a graph of the function by hand, correcting any errors appearing in the computer-generated graph. $$f(x)=\frac{4 x^{3}+4 x^{2}+7 x+4}{1+x^{2}}$$

Short answer

Question: Find the slant asymptote, vertical asymptotes, and sketch the graph of the function $$f(x)=\frac{4 x^{3}+4 x^{2}+7 x+4}{1+x^{2}}$$ along with the asymptotes. Answer: The slant asymptote of the function is $$y = 4x - 1$$. There are no vertical asymptotes. To sketch the graph, use a graphing utility to plot the function along with the slant asymptote and ensure that there are no vertical asymptotes.

Step-by-step solution

Polynomial Long Division

Using polynomial long division, divide the numerator $$4x^3 + 4x^2 + 7x +4$$ by the denominator $$1 + x^2$$. Doing so, we obtain: $$4x^3 + 4x^2 + 7x +4 = (1 + x^2)(4x - 1) +5$$ So the quotient is $$4x - 1$$, and this is the equation of the slant asymptote.

Find Vertical Asymptotes

To find the vertical asymptotes, we need to determine where the denominator of f(x) equals zero: $$1 + x^2 = 0$$ $$x^2 = -1$$ Since there are no real solutions to this equation, there are no vertical asymptotes for the function.

Graphing f(x) and Asymptotes

Using a graphing utility, plot the function $$f(x)=\frac{4 x^{3}+4 x^{2}+7 x+4}{1+x^{2}}$$ along with the slant asymptote $$y=4x - 1$$.

Hand Sketch and Corrections

Although not presented here, sketch the graph of the function according to the graphing utility output, but correct any errors that may have appeared in the computer-generated graph. Ensure that the graph matches the slant asymptote and does not display any vertical asymptotes, as these were not found in Step 2.

Key concepts

Polynomial Long Division

Polynomial long division is a method used to divide polynomials, similar to how traditional long division works with numbers. In essence, you divide the highest degree term in the numerator by the highest degree term in the denominator and then continue the process with the resulting remainders.
In this exercise, to find the slant asymptote of the function \(f(x) = \frac{4x^3 + 4x^2 + 7x + 4}{1 + x^2}\), we divide the numerator by the denominator using this method. The quotient we find during this division, \(4x - 1\), represents the slant asymptote, which is simply a line that the graph of \(f(x)\) will approach but never touch as \(x\) becomes very large or very small.
Steps to perform polynomial long division include:

• Identify the highest degree term in the numerator and the denominator.
• Divide the leading term of the numerator by the leading term of the denominator.
• Multiply the entire divisor by the result and subtract from the original polynomial.
• Repeat the process with the new polynomial formed as the remainder, until the remainder is of a lower degree than the divisor.

Understanding how to perform polynomial long division provides valuable insight into the behavior of rational functions, especially when identifying asymptotes.

Vertical Asymptotes

Vertical asymptotes are lines \(x = a\) where a function \(f(x)\) tends to infinity. They occur where the denominator of a rational function is zero, provided that the numerator isn’t zero at the same point.
For this function, we set the denominator \(1 + x^2 = 0\) to find potential vertical asymptotes. Solving \(x^2 = -1\), we see that there are no real solutions. This step confirms that there are no real number solutions, meaning \(f(x)\) does not have any vertical asymptotes.
This is because the expression \(x^2 = -1\) only has imaginary solutions \(x = i\) and \(x = -i\), which aren’t applicable in the context of real-number graphing. Thus, when considering real-valued functions like this one, it’s crucial to note the absence of vertical asymptotes.

Graphing Utility

A graphing utility is an electronic tool or software that helps in visualizing mathematical functions and their behaviors. It can quickly generate the graph of a function, including any asymptotes and critical points.
To properly visualize the function \(f(x) = \frac{4x^3 + 4x^2 + 7x + 4}{1 + x^2}\) and its asymptotes, you can use a graphing utility. By inputting the function and its slant asymptote \(y = 4x - 1\), you can observe how the graph behaves and verify if it aligns with theoretical expectations.
When using a graphing utility:

• Input the function correctly, paying attention to fractions and operators.
• Check if the software has the capability to plot asymptotes; manually add them if needed.
• Look for any discrepancies or errors the utility might make, such as improperly drawn asymptotes or failing to exclude non-existent features like vertical asymptotes.

The graph you produce will give a visual confirmation of the calculated asymptotes and help in understanding the function’s behavior.

Hand Sketch Corrections

After generating a graph using a graphing utility, it is beneficial to create a hand sketch of the function. This step lets you manually adjust any inaccuracies or clarify the behavior of the function around important features like asymptotes.
When sketching by hand, keep the following tips in mind:

• Note where the function approaches the slant asymptote \(y = 4x - 1\) but never touches it.
• Double-check that there are no vertical asymptotes on your graph, as confirmed by earlier calculations.
• Ensure your graph reflects the behavior around important points, like where the function crosses axes or changes direction.

This process of manual sketching helps reinforce your understanding of the function's behavior and attributes, providing a more complete picture than digital utilities alone. Regular practice in sketching enhances both problem-solving skills and familiarity with different types of functions.