Problem 58 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Slant asymptote: \(y = x+2\); Vertical asymptote: \(x=2\).

Step 1: Identify the degree of the numerator and denominator

The function given is \(r(x) = \frac{x^2}{x-2}\). The degree of the numerator \(x^2\) is 2, and the degree of the denominator \(x-2\) is 1. Since the degree of the numerator is one higher than the degree of the denominator, there is a slant asymptote.

Step 2: Find the slant asymptote using polynomial division

To find the slant asymptote, divide the numerator \(x^2\) by the denominator \(x-2\) using polynomial long division.ewline1. Divide: \(x^2 \div x = x\).ewline2. Multiply: \(x \times (x-2) = x^2 - 2x\).ewline3. Subtract: \((x^2 - 0x) - (x^2 - 2x) = 2x\).ewline4. Divide: \(2x \div x = 2\).ewline5. Multiply: \(2 \times (x-2) = 2x - 4\).ewline6. Subtract: \(2x - (2x - 4) = 4\).ewlineThe quotient is \(x + 2\), which is the slant asymptote \(y = x + 2\).

Step 3: Find the vertical asymptote

The vertical asymptote occurs where the denominator is zero. Set \(x-2 = 0\), which gives \(x = 2\). Thus, there is a vertical asymptote at \(x = 2\).

Step 4: Sketch the graph considering asymptotes

To sketch the graph, plot the slant asymptote \(y = x + 2\) as a dashed line and the vertical asymptote \(x = 2\) as another dashed line. Note that the function \(r(x)\) behaves like the slant asymptote when \(x\) moves towards positive or negative infinity. Approaching the vertical asymptote (\(x = 2\)), the function will approach infinity or negative infinity, depending on the side of approach.

Key Concepts

Slant Asymptote

A slant asymptote, also known as an oblique asymptote, occurs in rational functions when the degree of the numerator is exactly one degree higher than the degree of the denominator. In the function \( r(x) = \frac{x^2}{x-2} \), the numerator \( x^2 \) is of degree 2 and the denominator \( x-2 \) is of degree 1. This setup indicates the presence of a slant asymptote.
This asymptote can be found through polynomial long division of the function's numerator by its denominator. Following the division, the quotient \( x + 2 \) is revealed, which defines the line \( y = x + 2 \). This line is the slant asymptote, suggesting that as \( x \) approaches infinity or negative infinity, the function \( r(x) \) will closely follow this straight line.

Vertical Asymptote

Vertical asymptotes are vertical lines that the graph of a function approaches but never touches. They occur where the function is undefined, typically when the denominator of a rational function equals zero. For the function \( r(x) = \frac{x^2}{x-2} \), set the denominator equal to zero to find the vertical asymptote:
\( x - 2 = 0 \)
Solving this equation, we get \( x = 2 \). Hence, there's a vertical asymptote at \( x = 2 \), meaning the graph of \( r(x) \) will approach this line and tend towards positive or negative infinity as it nears the asymptote from either side.

Polynomial Division

Polynomial division is a crucial mathematical technique used to simplify or find asymptotes for certain rational functions. In this case, to locate the slant asymptote for \( r(x) = \frac{x^2}{x-2} \), we perform polynomial long division of the numerator \( x^2 \) by the denominator \( x-2 \).

• Start by dividing the leading term of the numerator \( x^2 \) by the leading term of the denominator \( x \) to get \( x \).
• Multiply \( x \) by the entire divisor \( x-2 \), resulting in \( x^2 - 2x \), and subtract this from the original numerator.
• You are left with \( 2x \). Divide this by \( x \) to obtain \( 2 \).
• Then multiply \( 2 \) by \( x-2 \) to get \( 2x - 4 \), subtract to find the remainder, which here is 4.

This results in the equation \( x + 2 \), defining our slant asymptote: \( y = x + 2 \).

Graph Sketching

Graph sketching involves detailed analysis of a function's behavior, particularly focusing on asymptotic tendencies. For \( r(x) = \frac{x^2}{x-2} \), after identifying its asymptotes, you begin sketching:

• First, draw the slant asymptote \( y = x + 2 \) as a dashed line to indicate the behavior at infinity.
• Next, draw the vertical asymptote \( x = 2 \) as another dashed line, where the function will diverge sharply.

Using these lines as guides, you'll notice that the function graph approaches these asymptotes. As \( x \) draws near the vertical asymptote, the graph veers sharply upwards or downwards, depending on the approach direction. When moving far left or right, \( r(x) \) aligns increasingly with the slant asymptote. This comprehensive layout helps visualize and better understand the function dynamics, crucial in detailed analysis and interpretation.