Problem 22 - 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. $$f(x)=\frac{1}{x}-2$$

Step-by-Step Solution

Answer

Shift the hyperbola down 2 units, domain: \(x \neq 0\), range: \(y \neq -2\).

Step 1: Understand the Parent Function

The parent function for this exercise is \(y = \frac{1}{x}\). This hyperbola has two branches that hug the x-axis and y-axis, with vertical and horizontal asymptotes at \(x = 0\) and \(y = 0\), respectively. Understanding its shape helps us visualize the transformations applied to it.

Step 2: Determine the Transformation

The function given is \(f(x) = \frac{1}{x} - 2\). This represents a vertical shift downward by 2 units from the graph of \(y = \frac{1}{x}\). Hence, the asymptote that was originally at \(y = 0\) is now at \(y = -2\).

Step 3: Sketch the Transformed Graph

To sketch the graph, start by drawing the basic hyperbola of \(y = \frac{1}{x}\). Then, shift the entire graph down by 2 units. Ensure the vertical asymptote remains at \(x = 0\) but the horizontal asymptote is now at \(y = -2\).

Step 4: Use a Graphing Calculator

Input \(f(x) = \frac{1}{x} - 2\) into a graphing calculator to see an accurate depiction of the graph. Observe the transformation confirms the vertical shift downward and check if the asymptotes are correctly positioned.

Step 5: Determine Domain and Range

The domain of \(f(x) = \frac{1}{x} - 2\) is all real numbers except \(x = 0\) (where the function is undefined), written as \(x eq 0\). The range consists of all real numbers except \(y = -2\) (the new horizontal asymptote), written as \(y eq -2\).

Key Concepts

Hyperbola

A hyperbola is a type of curve on a graph that looks like two mirrored arcs facing away from each other. A basic hyperbola, such as the graph of the function \( y = \frac{1}{x} \), has two parts, or branches, that approach the x-axis and y-axis,but never touch them.These axes are called asymptotes when they act as boundaries that the graph gets very close to, but never crosses.In the parent hyperbola \( y = \frac{1}{x} \):

• The vertical asymptote is at \( x = 0 \)
• The horizontal asymptote is at \( y = 0 \)

The hyperbola shows that the function is undefined at \( x = 0 \).Understanding the shape and behavior of the parent hyperbola is crucial before applying any transformations.

Vertical Shift

A vertical shift occurs when every point on a graph moves up or down by a certain number of units. For instance, if we alter \( y = \frac{1}{x} \) by subtracting 2, resulting in \( f(x) = \frac{1}{x} - 2 \), we perform a vertical shift downward by 2 units.In this transformation:

• The vertical asymptote remains unchanged at \( x = 0 \)
• The horizontal asymptote moves from \( y = 0 \) to \( y = -2 \)

This kind of shift does not affect the shape of the graphbut changes the position of the horizontal asymptote and subsequently,changes the range of the function.

Domain and Range

Understanding the domain and range of a function is essential in defining its behavior. For the function \( f(x) = \frac{1}{x} - 2 \), the domain consists of all the values \( x \) can take. However, since dividing by zero is undefined,\( x \) cannot be zero, thus the domain is all real numbers except \( x = 0 \).This means we write: \( x eq 0 \).For the range, which includes all the possible values of \( y \), any horizontal line drawn to intersect the graph will not touch the asymptote.Since the horizontal asymptote is at \( y = -2 \), the function's range is all real numbers except \( y = -2 \).Thus, it is represented as: \( y eq -2 \). This understanding helps us to see where the graph of the function exists on the coordinate plane.

Graphing Calculator

Using a graphing calculator is a helpful way to visualize complex functions, like hyperbolas.When you put \( f(x) = \frac{1}{x} - 2 \) into the calculator, it automatically generates a graph,allowing you to observe the transformation clearly.Check these points while using the graphing calculator:

• Verify the vertical asymptote is still at \( x = 0 \)
• Ensure that the horizontal asymptote has shifted to \( y = -2 \)

This immediate and accurate depiction aids in confirming what the manual calculations predict.Additionally, graphing calculators can provide detailed numerical data about the function,supporting a deeper understanding of its characteristics and helping verify its domain and range.