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 of its asymptotes with a graphing utility. Then sketch a graph of the function by hand, correcting any errors appearing in the computer-generated graph. $$f(x)=\frac{3 x^{2}-2 x+5}{3 x+4}$$

Short answer

Answer: There is no horizontal asymptote, as the function has a slant asymptote given by \(y = x-\frac{4}{3}\). The vertical asymptote is at \(x = -\frac{4}{3}\).

Step-by-step solution

Perform Polynomial Long Division

In order to find the slant asymptote, we need to perform polynomial long division. Divide \(3x^{2}-2x+5\) by \(3x+4\): $$ \require{enclose} \begin{array}{r@{\hskip2pt}rc@{\hskip1pt}} \multicolumn{2}{r}{x} & -\dfrac{4}{3} \\ \cline{2-3} 3x+4 & \enclose{longdiv}{3x^2 - 2x+5} \\ \cline{2-2} \multicolumn{2}{r}{-3x^2 - 4x} \\ \cline{2-2} \multicolumn{2}{r}{2x + 5} \\ \cline{2-2} \multicolumn{2}{r}{2x + \dfrac{8}{3}} \\ \cline{2-2} \multicolumn{2}{r}{-\dfrac{7}{3}} \\ \end{array} $$ Thus, \(f(x) = x - \frac{4}{3} + \frac{-\frac{7}{3}}{3x+4}\). The slant asymptote is the quotient part, which is \(y = x - \frac{4}{3}\).

Find Vertical Asymptotes

To find the vertical asymptotes, we need to set the denominator of the function equal to zero and solve for \(x\).So, \(3x + 4 = 0\): $$x = -\dfrac{4}{3}$$ The function has one vertical asymptote at \(x=-\frac{4}{3}\).

Graph the Function and Asymptotes

Now, use a graphing software to draw the function \(f(x) = \frac{3x^2 - 2x + 5}{3x + 4}\) and its asymptotes: 1. Slant asymptote: \(y=x-\frac{4}{3}\). 2. Vertical asymptote: \(x=-\frac{4}{3}\). After generating the graph, sketch the graph by hand and correct any errors if necessary. Ensure that the function conforms to its asymptotes for large values of \(x\).

Key concepts

Polynomial Long Division

Polynomial long division is a method used for dividing one polynomial by another, drawing parallels to long division with numbers. Imagine you want to divide a more complex polynomial by a simpler one, like \(3x^{2}-2x+5\) by \(3x+4\). Here’s how this process works:

• Start by dividing the leading term of the dividend (\(3x^2\)) by the leading term of the divisor (\(3x\)). This gives you \(x\), the first term of the quotient.
• Multiply the entire divisor by this term and subtract the result from the original polynomial.
• Bring down the next term from the dividend and repeat the process until you've worked through all terms.

For our function, following these steps results in a quotient of \(x - \frac{4}{3}\) and a remainder. The important part for slant asymptotes is the linear quotient part.

Slant Asymptote

Slant asymptotes, or oblique asymptotes, occur in rational functions when the degree of the polynomial in the numerator is one higher than in the denominator. They provide a linear approximation of the function as \(x\) approaches positive or negative infinity.
In our function \(f(x) = \frac{3x^2 - 2x + 5}{3x + 4}\), using polynomial long division, we discovered that the linear part of the quotient \(y = x - \frac{4}{3}\) represents the slant asymptote. This line acts as a guide that the graph of the function will approximate, but not touch as \(x\) becomes very large or very small.

• Slant asymptotes differ from horizontal asymptotes, which occur when the degrees of the numerator and denominator are equal.
• Unlike horizontal asymptotes, which can be leveled, slant asymptotes "tilt," providing a direction of the graph.
• The graph of \(f(x)\) will tend towards this line but will never intersect it as it extends along the x-axis.

Vertical Asymptote

Vertical asymptotes occur in rational functions where the denominator equals zero, leading to values approaching infinity. The graph approaches these asymptotes but never actually reaches them, creating a discontinuity.
To find the vertical asymptote for our function \(f(x) = \frac{3x^2 - 2x + 5}{3x + 4}\), we set the denominator equal to zero: \(3x + 4 = 0\). Solving this equation, we find \(x = -\frac{4}{3}\). Here:

• The line \(x = -\frac{4}{3}\) is a vertical asymptote.
• The graph of \(f(x)\) approaches this line but will never cross it, indicating a division in the behavior of the function.
• As \(x\) approaches \(x = -\frac{4}{3}\) from either side, \(f(x)\) tends towards infinity or negative infinity, depending on the direction.

Graphically, understanding vertical asymptotes helps in sketching rational functions as it indicates a clear division point where the function "breaks."