Problem 35 - College Algebra

Question

For the following exercises, use the given transformation to graph the function. Note the vertical and horizontal asymptotes. The reciprocal function shifted up two units.

Step-by-Step Solution

Answer

The reciprocal function shifted up two units is \( f(x) = \frac{1}{x} + 2 \), with vertical asymptote at \( x = 0 \) and horizontal asymptote at \( y = 2 \).

Step 1: Understand the Basic Function

The reciprocal function is given by \( f(x) = \frac{1}{x} \). The graph of this function has a vertical asymptote at \( x = 0 \) and a horizontal asymptote at \( y = 0 \).

Step 2: Apply the Transformation

According to the problem, the reciprocal function needs to be shifted up by 2 units. This transformation changes the function to \( f(x) = \frac{1}{x} + 2 \).

Step 3: Identify the New Horizontal Asymptote

Shifting the function up by 2 units also shifts the horizontal asymptote from \( y = 0 \) to \( y = 2 \). This is because the value of the function will approach 2 as \( x \) goes to infinity or negative infinity.

Step 4: Determine the Vertical Asymptote

The vertical asymptote remains unchanged by vertical shifts. Therefore, the vertical asymptote of \( f(x) = \frac{1}{x} + 2 \) is still at \( x = 0 \).

Step 5: Graph the Function

To graph the function \( f(x) = \frac{1}{x} + 2 \), plot the shifted function, ensuring the graph is centered on the vertical line \( x = 0 \) and approaches the horizontal line \( y = 2 \) at large values of \( x \) and as \( x \) approaches zero from either direction.

Key Concepts

Reciprocal Function

The reciprocal function is a basic yet important mathematical concept often encountered in algebra and calculus. It is represented by the formula \( f(x) = \frac{1}{x} \). When graphed, the reciprocal function creates a hyperbola with two distinct branches.

• Characteristics: The function is defined for all real numbers except zero, as division by zero is undefined.
• Graph Shape: The graph consists of two symmetrical branches located in the first and third quadrants of the Cartesian plane.
• Behavior: As the value of \( x \) increases or decreases, the function values approach zero, but never actually reach it.

Graph transformations can adjust the placement of the reciprocal function, such as translating it vertically or horizontally, affecting its asymptotes - lines that the graph will approach but never touch or cross.

Vertical Asymptote

Vertical asymptotes are lines parallel to the y-axis that a graph approaches but never intersects. For the reciprocal function \( f(x) = \frac{1}{x} \), the vertical asymptote is found at \( x = 0 \). This is because at \( x = 0 \), the function is undefined, leading the graph to rise or fall indefinitely.

• Indication: Vertical asymptotes show the limitations within a function, often indicating discontinuities.
• Behavior: As \( x \) gets very close to the asymptote, the function’s value will either increase to positive infinity or decrease to negative infinity, depending on the direction of approach.

Importantly, shifting a function vertically (up or down) does not affect its vertical asymptote. This is why shifting \( f(x) = \frac{1}{x} \) up by 2 units to \( f(x) = \frac{1}{x} + 2 \) keeps the asymptote steadily at \( x = 0 \).

Horizontal Asymptote

Horizontal asymptotes are lines parallel to the x-axis that a graph approaches as \( x \) moves towards infinity or negative infinity. For the basic reciprocal function \( f(x) = \frac{1}{x} \), the horizontal line \( y = 0 \) acts as such an asymptote.

• Function Values: As \( x \) increases or decreases significantly, the output value \( f(x) \) nears this asymptote.
• Impact of Transformations: A vertical shift in the function, such as moving it up by 2 units resulting in \( f(x) = \frac{1}{x} + 2 \), repositions the horizontal asymptote.

For \( f(x) = \frac{1}{x} + 2 \), the horizontal asymptote becomes \( y = 2 \). This occurs because, as \( x \) moves further from zero, the value of \( f(x) \) approaches 2 instead of 0, effectively raising the graph. Understanding how such transformations impact the reciprocal function's asymptotes is key to mastering graph-related problems.