Problem 52 - Calculus A New Horizon

Question

Sketch the graph of $$y=\frac{1}{(x-a)(x-b)}$$ assuming that \(a \neq b\)

Step-by-Step Solution

Answer

Sketch the graph with vertical asymptotes at \(x = a\) and \(x = b\), a horizontal asymptote at \(y = 0\), and a y-intercept at \(0, \frac{1}{ab}\).

Step 1: Understand the function components

The function is given by \(y = \frac{1}{(x-a)(x-b)}\). It has a numerator of 1 and a denominator of \((x-a)(x-b)\). The critical points are where the function is undefined, which occurs at \(x = a\) and \(x = b\), indicating vertical asymptotes.

Step 2: Find intercepts

To find the y-intercept, set \(x = 0\). Substitute into the function: \(y = \frac{1}{(0-a)(0-b)} = \frac{1}{ab}\). Thus, the y-intercept is at \( (0, \frac{1}{ab}) \). There is no x-intercept as the numerator is constant (1) and cannot be zero.

Step 3: Determine asymptotic behavior

The vertical asymptotes are located at \(x = a\) and \(x = b\) because the function is undefined at these points. To find horizontal asymptotes, consider the limits as \(x\) approaches infinity or negative infinity. Since the degree of the denominator is greater than that of the numerator, \(y \to 0\) as \(x\to\infty\) or \(x\to-\infty\). Therefore, \(y = 0\) is a horizontal asymptote.

Step 4: Analyze the sign of function in intervals

Examine the sign of \(y = \frac{1}{(x-a)(x-b)}\) in intervals determined by the asymptotes (\((-\infty, a)\), \((a, b)\), \((b, \infty)\)). Pick test points in each interval, calculate \(y\), and determine if the values are positive or negative, which will indicate the direction of the curve relative to the x-axis.

Step 5: Consider behavior near asymptotes

As \(x\) approaches \(a\) from the left, \(y\) becomes very negative, and from the right, \(y\) becomes very positive. Conversely, as \(x\) approaches \(b\) from the left, \(y\) becomes very positive, and from the right, \(y\) becomes very negative. This helps determine how the graph approaches the vertical asymptotes.

Step 6: Sketch the graph

Plot the key features: vertical asymptotes at \(x = a\) and \(x = b\), and horizontal asymptote at \(y = 0\). The y-intercept occurs at \((0, \frac{1}{ab})\). Using the sign analysis in each interval, sketch the graph piecewise, noting that it crosses or touches the horizontal asymptote only at extreme x-values.

Key Concepts

Asymptotic Behavior

Rational functions, such as \( y = \frac{1}{(x-a)(x-b)} \), often exhibit asymptotic behavior, showcasing how the graph behaves near certain lines known as asymptotes.

Types of Asymptotes:

• **Vertical Asymptotes**: These occur due to division by zero, indicating points where the function is undefined. In our function, vertical asymptotes appear at \( x = a \) and \( x = b \). As \( x \) approaches these values, \( y \) approaches infinity or negative infinity.
• **Horizontal Asymptotes**: These appear as the graph approaches a certain value as \( x \) tends towards infinity. For our function, since the degree of the denominator is higher than the numerator, \( y \) approaches 0 as \( x \) moves towards \( \pm \infty \).

Understanding the asymptotic behavior not only highlights where the function may be undefined but also indicates important limits and behaviors at extreme values of \( x \).

These concepts help us predict and sketch how the graph behaves near its boundaries.

Graph Sketching

Graph sketching for rational functions involves determining how the graph is shaped by its asymptotes and intercepts.

Here's a simplified approach for our function:

• **Identify Vertical Asymptotes**: Mark \( x = a \) and \( x = b \) since the function is undefined at these points.
• **Determine Horizontal Asymptote**: Recognize \( y = 0 \) as the graph tends toward this line as \( x \) moves toward infinity.
• **Evaluate Sign Changes**: Divide the x-axis into intervals \( (-\infty, a) \), \( (a, b) \), and \( (b, \infty) \). Analyze how the function's values change in each interval by selecting test points; for example, choose \( x = c < a \), \( c_2 \) between \( a \) and \( b \), and \( c > b \).

By piecing together how the function behaves in these intervals, including its approach toward asymptotes, you can create a complete graph.

Remember, rational functions might not touch or intersect the horizontal asymptote except at very large values of \( x \). This is significant because it frames the graph's curvature.

Intercepts in Functions

Intercepts provide crucial anchor points for analyzing the overall shape and position of rational functions on the coordinate plane.

**Finding Intercepts:**

• **Y-Intercepts**: Set \( x = 0 \) to determine where the curve intersects the y-axis. For \( y = \frac{1}{(x-a)(x-b)} \), substituting \( x = 0 \) gives the y-intercept at \( \left(0, \frac{1}{ab}\right) \).
• **X-Intercepts**: Typically found by solving \( y = 0 \). However, because the numerator of our function is a constant 1, it never equals zero, meaning no x-intercepts exist.

In summary, the role of intercepts helps provide a framework for sketching the graph by identifying these critical intersections where the curve crosses the axes. In our rational function, since the y-intercept provides one main point of intersection, this unique property necessitates other considerations to comprehend its shape fully.