Problem 59 - Precalculus

Question

Apply a graphing utility to graph \(y_{1}=\frac{5 x+4}{x^{2}+x-2}\) and \(y_{2}=\frac{3}{x-1}+\frac{2}{x+2}\) in the same viewing rectangle. Is \(y_{2}\) the partial-fraction decomposition of \(y_{1} ?\)

Step-by-Step Solution

Answer

Yes, \( y_{2} = \frac{3}{x-1} + \frac{2}{x+2} \) matches the partial-fraction decomposition of \( y_{1} = \frac{5x + 4}{x^{2} + x - 2} \).

Step 1: Set up the equations for graphing

We need to prepare both functions for graphing using a graphing utility like Desmos or a graphing calculator. Function 1 is \( y_{1} = \frac{5x + 4}{x^{2} + x - 2} \) and function 2 is \( y_{2} = \frac{3}{x-1} + \frac{2}{x+2} \). These are the equations you will input into your graphing utility.

Step 2: Input Function \( y_{1} \)

Enter the first function into your graphing utility. The function \( y_{1} = \frac{5x + 4}{x^{2} + x - 2} \) should be plotted. Observe any features like asymptotes, intercepts, and general shape.

Step 3: Input Function \( y_{2} \)

Enter the second function into the graphing utility. The function \( y_{2} = \frac{3}{x-1} + \frac{2}{x+2} \) will be plotted alongside \( y_{1} \). Watch for similar features relative to \( y_{1} \).

Step 4: Analyze the Graphs

Examine both graphs to determine if they coincide over their domains. Look specifically at how they approach any vertical asymptotes, horizontal asymptotes, and their general curves. They should be identical if \( y_{2} \) is the partial fraction decomposition of \( y_{1} \).

Step 5: Verify with Algebraic Decomposition

Perform algebraic verification by ensuring \( y_{2} = y_{1} \). For \( y_{1} \), factorize the denominator: \( x^2 + x - 2 = (x-1)(x+2) \). The numerator 5x + 4 can be decomposed as the sum of fractions with these denominators, consistent with \( y_{2} \). Confirm that the algebraic manipulation aligns with the graph.

Key Concepts

Graphing Utility

A graphing utility, such as Desmos or a graphing calculator, is a tool that allows you to plot mathematical functions to visually analyze their behaviors. These tools are essential for visualizing complex functions like rational functions.

Using a graphing utility, you can:

• Plot multiple functions on the same graph for comparison.
• Identify and analyze points of interest such as intercepts and asymptotes.
• Visually verify mathematical solutions, like partial fraction decompositions.

When using a graphing utility, ensure accuracy by inputting the function correctly. Small errors can significantly impact the graph's representation. Adjust the viewing window to include essential features and ensure both functions, like our example functions, fit within the same frame for comparison.

Rational Functions

Rational functions are fractions of polynomials. They take the general form \( R(x) = \frac{P(x)}{Q(x)} \), where \(P(x)\) and \(Q(x)\) are polynomials. The function \( y_{1} = \frac{5x + 4}{x^2 + x - 2} \) is an example of a rational function.

Rational functions can:

• Display complex behaviors including asymptotes and intercepts.
• Be broken down into simpler parts through partial fraction decomposition, particularly valuable in calculus and algebra.

In our case, the function \( y_1 \) is decomposed into \( y_2 = \frac{3}{x-1} + \frac{2}{x+2} \). These simpler rational components make the function easier to analyze and integrate mathematically. Each partial fraction corresponds to factors of the original function's denominator.

Asymptotes

Asymptotes are lines that a graph approaches but never touches. They can provide insight into the behavior of rational functions like \( y_{1} \) and \( y_{2} \). There are typically two types of asymptotes associated with rational functions: horizontal and vertical.

### Vertical AsymptotesThese occur when the denominator of the function equals zero, creating points where the graph goes to infinity. For \( y_1 = \frac{5x + 4}{x^2 + x - 2} \), set \(x^2 + x - 2 = 0\) to find the vertical asymptotes.

• The solutions to \((x-1)(x+2) = 0\) are \(x = 1\) and \(x = -2\).
• These values correspond to the asymptotes shared by \( y_2 \), making them crucial points of interest for graphing.

### Horizontal AsymptotesThese are determined by the degrees of the polynomial in the numerator and denominator. If the degree of the numerator is less than or equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients. For \( y_1 \), the degree of the numerator is less than the denominator, so the horizontal asymptote is \( y = 0 \).Understanding asymptotes helps to predict how the graph of a rational function behaves as \(x\) approaches infinity or the vertical boundaries.