Problem 17 - Calculus - AP Edition

Question

Show that \(y=x+3\) is an oblique asymptote of the graph of \(f(x)=x^{2} /(x-3) .\) Sketch the graph of \(y=f(x)\) showing this asymptotic behavior.

Step-by-Step Solution

Answer

The oblique asymptote of \(f(x)=x^2/(x-3)\) is \(y=x+3\).

Step 1: Define Oblique Asymptote

An oblique asymptote, also known as a slant asymptote, occurs for rational functions when the degree of the numerator is exactly one more than the degree of the denominator. In such cases, the oblique asymptote can be found by performing polynomial long division of the numerator by the denominator.

Step 2: Perform Polynomial Long Division

Divide the numerator, \(x^2\), by the denominator, \(x-3\), to find the oblique asymptote. 1. Divide the leading term of the numerator, \(x^2\), by the leading term of the denominator, \(x\), which gives \(x\).2. Multiply \(x\) by \(x-3\) to get \(x^2 - 3x\).3. Subtract \(x^2 - 3x\) from \(x^2\) to get \(3x\).4. Divide \(3x\) by \(x\) to get \(+3\).5. Multiply \(3\) by \(x-3\) to get \(3x - 9\).6. Subtract \(3x - 9\) from \(3x\) to get \(+9\).The quotient is \(x+3\), which is the equation of the oblique asymptote.

Step 3: Verify the Oblique Asymptote

Verify that as \(x\) approaches infinity, the remainder \((+9)/(x-3)\) tends to zero, confirming that the asymptote of \(f(x) = x^2/(x-3)\) is indeed \(y = x + 3\). As \(x \to \infty\) or \(x \to -\infty\), the term \(\frac{9}{x-3}\) approaches 0, so the graph of \(f(x)\) approaches the line \(y = x + 3\).

Step 4: Sketch the Graph

1. Draw the coordinate axes.2. Sketch the line \(y = x + 3\) by plotting points and drawing a straight line through them.3. Note the vertical asymptote at \(x = 3\) because the denominator of \(f(x)\) is zero at this point.4. Sketch the graph of \(f(x) = \frac{x^2}{x-3}\), showing it approaching \(y = x + 3\) asymptotically away from the vertical asymptote.

Key Concepts

Rational Functions

Understanding rational functions is essential to comprehend concepts like asymptotes. A rational function is a function that can be expressed as the quotient of two polynomials. Specifically, it takes the form \( f(x) = \frac{P(x)}{Q(x)} \), where both \(P(x)\) and \(Q(x)\) are polynomials. The behavior of a rational function is largely determined by the degrees of these polynomials:

• If the degree of the polynomial in the numerator is higher, the function can have an oblique or horizontal asymptote.
• If the degree in the denominator is higher, the rational function's graph will approach the x-axis (y = 0) as an asymptote.
• When the denominator is zero, vertical asymptotes occur, since the function will be undefined at those points.

For instance, in the function \( f(x) = \frac{x^2}{x-3} \), the degree of the numerator is one greater than that of the denominator, which leads us to expect an oblique asymptote in this scenario.

Polynomial Long Division

Polynomial long division is a method used to simplify the expression of a rational function, especially useful for finding oblique asymptotes. This process works similarly to numerical long division. You iterate through the following steps:

• Divide the leading term of the numerator by the leading term of the denominator.
• Multiply the entire divisor by this result and subtract it from the original polynomial.
• Repeat the process with the new polynomial obtained after subtraction.

Applying polynomial long division to \( \frac{x^2}{x-3} \) provides a quotient of \( x+3 \). This calculates our oblique asymptote. The remainder, \( \frac{9}{x-3} \), approaches zero as \( x \) tends to infinity, confirming the asymptotic nature along the line \( y = x + 3 \). Polynomial long division thus helps in breaking down complex functions into simpler asymptotic behaviors.

Graph Sketching

Graph sketching involves drawing the function's curve, highlighting key characteristics such as intercepts and asymptotes. When graphing \( f(x) = \frac{x^2}{x-3} \), certain aspects need consideration:

• Draw the axes: It's crucial to establish a graph framework by plotting coordinate axes.
• Identify and draw asymptotes: For \( f(x) \), there is a vertical asymptote at \( x = 3 \), and an oblique asymptote at \( y = x + 3 \).
• Plot key points: Plot points around the intercepts and asymptotes to accurately represent graph behavior.
• Sketch the behavior: Near \( x = 3 \), the function goes to infinity, while far away from \( x = 3 \), it approaches \( y = x + 3 \).

By following these steps, you accommodate all significant features of the rational function, providing a clear representation of its behavior across different parts of the graph.