Problem 20 - A Graphical Approach to College Algebra

Question

Explain how the graph of \(f\) can be obtained from the graph of \(y=\frac{1}{x}\) or \(y=\frac{1}{x^{2}}\) Draw a sketch of the graph of \(f\) by hand. Then generate an accurate depiction of the graph with a graphing calculator. Finally, give the domain and range. $$f(x)=\frac{1}{x-3}$$

Step-by-Step Solution

Answer

The graph of \( f(x) = \frac{1}{x-3} \) is obtained by shifting \( y = \frac{1}{x} \) 3 units right, with domain \( x \neq 3 \) and range \( y \neq 0 \).

Step 1: Identify the Base Graph

The given function is based on the form \( y = \frac{1}{x} \). The base graph for this function is the hyperbola represented by \( y = \frac{1}{x} \). This graph has vertical and horizontal asymptotes at \( x = 0 \) and \( y = 0 \) respectively.

Step 2: Determine Transformations

The function \( f(x) = \frac{1}{x-3} \) can be rewritten as \( y = \frac{1}{(x-3)} \). This transformation involves a horizontal shift to the right by 3 units, as it modifies \( x \) to \( x-3 \), shifting the asymptotes to \( x=3 \) and \( y=0 \).

Step 3: Sketch the Graph

To sketch the graph, start by drawing the horizontal asymptote at \( y = 0 \). Then, draw the vertical asymptote at \( x = 3 \). The hyperbolic graph will approach these asymptotes but never touch them. It will lie in the first and third quadrants due to the positive nature of \( y = \frac{1}{x} \).

Step 4: Use a Graphing Calculator

Using a graphing calculator, input \( y = \frac{1}{x-3} \). This should show a graph with the same shape as the base graph, but shifted 3 units to the right. The graph will closely match your sketch, confirming the location of the asymptotes.

Step 5: Determine the Domain and Range

For the function \( f(x) = \frac{1}{x-3} \), the domain excludes \( x = 3 \) where the function is undefined, so the domain is \( x \in (-\infty, 3) \cup (3, \infty) \). The range excludes \( y = 0 \) since the function has a horizontal asymptote at \( y = 0 \), so the range is \( y \in (-\infty, 0) \cup (0, \infty) \).

Key Concepts

Asymptotes

Asymptotes are lines that a graph can get very close to but usually never touch. They help us understand the behavior of functions as they approach certain values. There are two main types of asymptotes relevant to the function \( f(x) = \frac{1}{x-3} \):

• Vertical Asymptotes: These occur where the function is undefined. For \( f(x) = \frac{1}{x-3} \), the function is undefined at \( x = 3 \) because dividing by zero is not possible. Thus, \( x = 3 \) is a vertical asymptote.
• Horizontal Asymptotes: These indicate the value that the function approaches as \( x \) goes to infinity or negative infinity. Here, the horizontal asymptote is \( y = 0 \). As \( x \) moves further away from 3, \( f(x) \) gets closer to 0 but never quite reaches it.

Asymptotes guide the sketching of the graph, indicating regions where the curve bends toward these invisible lines.

Domain and Range

The domain and range of a function tell us, respectively, which \( x \) values we can input and which \( y \) values we can get as output. For our function \( f(x) = \frac{1}{x-3} \):

• The domain includes all real numbers except where the function is undefined. Since the function is undefined at \( x = 3 \), the domain is \( x \in (-\infty, 3) \cup (3, \infty) \).
• The range includes all real \( y \) values except where the graph has a horizontal asymptote. Here, there is a horizontal asymptote at \( y = 0 \), so the range is \( y \in (-\infty, 0) \cup (0, \infty) \).

Understanding the domain and range helps us know where a graph can and cannot exist, enabling us to avoid undefined areas.

Hyperbolic Functions

Hyperbolic functions like \( y = \frac{1}{x} \) and its transformations have a distinct shape that resembles two branches approaching each other but never meeting. The function \( f(x) = \frac{1}{x-3} \) inherits this hyperbolic form but shifts horizontally.

• Basic Characteristics: Hyperbolic graphs often exist in two quadrants, either in the first and third or the second and fourth, depending on the transformation. For \( f(x) = \frac{1}{x-3} \), the branches are in the first and third quadrants, indicating that as \( x \) increases or decreases away from 3, \( f(x) \) approaches zero.
• Transformation: The transformation \( x \rightarrow x-3 \) shifts the entire graph to the right by 3 units, moving the vertical asymptote to \( x = 3 \).

Using these features helps us graph hyperbolic functions accurately and understand how they behave under various transformations.