Problem 17 - 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. $$f(x)=\frac{2}{x}$$

Step-by-Step Solution

Answer

The graph of \(f(x) = \frac{2}{x}\) is a vertical stretch of \(y = \frac{1}{x}\). Its domain and range are both \(x \neq 0\) and \(y \neq 0\), respectively.

Step 1: Understanding the Parent Function

The parent function is given as \(y = \frac{1}{x}\), which is also known as the reciprocal function. It is characterized by two branches in the first and third quadrants, with a vertical asymptote at \(x = 0\) and a horizontal asymptote at \(y = 0\).

Step 2: Applying a Vertical Stretch

The function \(f(x) = \frac{2}{x}\) can be obtained by applying a vertical stretch by a factor of 2 to the graph of \(y = \frac{1}{x}\). This transformation multiplies all y-values by 2, causing the graph to move further away from the x-axis while maintaining the same shape.

Step 3: Sketching the Transformed Graph

Begin by sketching the basic shape of \(y = \frac{1}{x}\), noting the branches in the first and third quadrants. Stretch each point vertically by a factor of 2 to represent \(f(x) = \frac{2}{x}\). Be sure both vertical (x = 0) and horizontal (y = 0) asymptotes remain unchanged.

Step 4: Graphing Calculator Verification

Use a graphing calculator to plot \(f(x) = \frac{2}{x}\). Make sure the graph matches your hand sketch, showing a steeper curve compared to \(y = \frac{1}{x}\) and respecting the same asymptotes.

Step 5: Determining the Domain and Range

The domain of \(f(x) = \frac{2}{x}\) is all real numbers except zero, \(x eq 0\), because division by zero is undefined. The range is all real numbers except zero, \(y eq 0\), as there is no value of \(x\) for which the output is zero.

Key Concepts

Reciprocal Function

In mathematics, the reciprocal function is one of the fundamental types of functions you will encounter. This function is described by the equation \( y = \frac{1}{x} \). Its graph exhibits a distinctive shape made up of two branches. These branches are located in the first and third quadrants of the Cartesian coordinate plane.

An important feature to note about the reciprocal function is its asymptotes. A vertical asymptote is present at \( x = 0 \), where the function is undefined because division by zero is impossible. Likewise, there is a horizontal asymptote at \( y = 0 \), indicating that the function values approach zero as \( x \) becomes very large or very small, but they never actually reach zero.

Vertical Stretch

A vertical stretch happens when you multiply the y-values of a function by a certain factor. For the function \( f(x) = \frac{2}{x} \), a vertical stretch by a factor of 2 is applied to the parent function \( y = \frac{1}{x} \). This transformation amplifies the y-values without altering the x-values.

Essentially, each point on the graph of \( y = \frac{1}{x} \) is "stretched" vertically upward or downward by the factor of 2. The graph moves away from the x-axis, maintaining its mirrored layout across the first and third quadrants. The key asymptotes at \( x = 0 \) and \( y = 0 \) remain the same, providing a guide to how the graph behaves for extreme values of \( x \).

Domain and Range

The domain of a function describes all the possible \( x \)-values, while the range describes the possible \( y \)-values. For \( f(x) = \frac{2}{x} \), the function is defined for all real numbers except at \( x = 0 \) because the function becomes undefined at zero. Thus, the domain of \( f(x) \) is all real numbers except zero, expressed as \( x eq 0 \).

Regarding the range, the function values cover all real numbers except zero. No matter the input, the output will never be exactly zero due to the horizontal asymptote at \( y = 0 \). Therefore, the range of \( f(x) \) is all real numbers excluding zero, noted as \( y eq 0 \). This detail ensures that we understand how the function's graph approaches but never reaches the x-axis.

Graphing Calculator Usage

Using a graphing calculator can be incredibly helpful when dealing with complex functions, like \( f(x) = \frac{2}{x} \). To start, input the function into the calculator to see its graphical representation. This tool provides a visual confirmation of what you might sketch by hand.

When you plot \( f(x) \) on a graphing calculator, check that the graph appears with steeper branches compared to the parent function \( y = \frac{1}{x} \), owing to the vertical stretch. Look at the position of the asymptotes to ensure accuracy. They should remain at \( x = 0 \) and \( y = 0 \). This method can give you confidence that your understanding and manual graphing are correct and provide a clear visual for study and interpretation.