Problem 68 - Algebra and Trigonometry

Question

Find the slant asymptote, the vertical asymptotes, and sketch a graph of the function. $$ r(x)=\frac{3 x-x^{2}}{2 x-2} $$

Step-by-Step Solution

Answer

The slant asymptote is \(y = -\frac{1}{2}x + 1\) and the vertical asymptote is \(x = 1\).

Step 1: Identify the Slant Asymptote

To find the slant asymptote, divide the numerator by the denominator using polynomial long division. Divide \(3x - x^2\) by \(2x - 2\) to obtain:1. Rewrite \(-x^2 + 3x\) for division.2. Divide the leading term \(-x^2\) by \(2x\) to get \(-\frac{1}{2}x\).3. Multiply \(-\frac{1}{2}x\) by \(2x - 2\) to get \(-x^2 + x\).4. Subtract \(-x^2 + x\) from \(-x^2 + 3x\), obtaining \(2x\).5. Divide \(2x\) by \(2x\) to get \(1\).6. Thus the division results in \(-\frac{1}{2}x + 1\).So, the slant asymptote is \(y = -\frac{1}{2}x + 1\).

Step 2: Identify the Vertical Asymptotes

Vertical asymptotes occur where the denominator equals zero, as long as the numerator does not simultaneously equal zero. Solve:\[2x - 2 = 0\]\[2x = 2\]\[x = 1\]So there is a vertical asymptote at \(x = 1\).

Step 3: Sketch the Graph

To sketch the graph:1. Plot the vertical asymptote line \(x = 1\).2. Plot the slant asymptote \(y = -\frac{1}{2}x + 1\).3. Determine additional points by substituting some values into \(r(x)\).4. Ensure the graph approaches these asymptotes as \(x\) moves towards extremes in both negative and positive directions.

Key Concepts

Vertical Asymptote

A vertical asymptote represents a value of \(x\) that makes the denominator of a rational function zero, leading to the function becoming undefined at that point. For the given function \(r(x) = \frac{3x - x^2}{2x - 2}\), we find vertical asymptotes by setting the denominator equal to zero:

• Start with the equation \(2x - 2 = 0\).
• Solve for \(x\) by adding 2 to both sides to get \(2x = 2\).
• Divide by 2 to find \(x = 1\).

This means there is a vertical asymptote at \(x = 1\). The graph of the function will never cross this line, and as \(x\) approaches 1 from either side, the value of \(r(x)\) will approach positive or negative infinity.

Polynomial Long Division

Polynomial long division is an essential method used to find the slant asymptote of a rational function where the degree of the numerator is one degree higher than the denominator. In our exercise, we aim to divide the polynomial \(-x^2 + 3x\) by \(2x - 2\).To carry out the division:

• Divide the leading term of the numerator \(-x^2\) by the leading term of the denominator \(2x\), resulting in \(-\frac{1}{2}x\).
• Multiply \(-\frac{1}{2}x\) by \(2x - 2\) to obtain \(-x^2 + x\).
• Subtract this from the original numerator \(-x^2 + 3x\) to result in \(2x\).
• Continue by dividing \(2x\) by \(2x\) to finish with 1.

Thus, the result of the division is \(-\frac{1}{2}x + 1\). It indicates the slant asymptote: \(y = -\frac{1}{2}x + 1\). This line illustrates the behavior as \(x\) grows significantly in positive or negative direction.

Graphing Rational Functions

Graphing rational functions involves plotting both the asymptotes and additional points to visualize the behavior of the function. Here are the steps we follow:

• Identify and draw the vertical asymptote \(x = 1\). This line is crucial as it represents a boundary the graph cannot cross.
• Plot the slant asymptote \(y = -\frac{1}{2}x + 1\). This line demonstrates how the graph behaves as \(x\) approaches extreme values, guiding the overall slope of the graph.
• Select additional points by plugging in \(x\) values into \(r(x)\) to obtain corresponding \(y\) values.
• Connect these points, making sure that the graph adheres closely to the asymptotes as \(x\) tends towards infinity or negative infinity.
• Verify the graph aligns with the mathematical rules of asymptotes, never crossing the vertical but getting infinitely close as needed.

These steps help cultivate a clear understanding of how to interpret and depict rational functions effectively.