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Chapter 4: Problem 17

Graph the rational function by applying transformations to the graph of \(y=\frac{1}{x}\). $$f(x)=\frac{1}{x-2}$$

Short Answer

Expert verified
Shift the graph of \( y = \frac{1}{x} \) 2 units to the right to graph \( f(x) = \frac{1}{x-2} \).

Step by step solution

01

Identify Parent Function

The parent function given is \( y = \frac{1}{x} \), which is a rational function known as the reciprocal function. Its graph is a hyperbola with vertical and horizontal asymptotes along the x-axis and y-axis, respectively.
02

Determine Transformations

The function given is \( f(x) = \frac{1}{x-2} \). We can see that this function involves a horizontal shift. The term \( x-2 \) indicates a horizontal shift to the right by 2 units.
03

Apply Horizontal Shift

To graph \( f(x) = \frac{1}{x-2} \), apply the horizontal shift to the graph of \( y = \frac{1}{x} \). Move every point on the hyperbola 2 units to the right. The vertical asymptote, originally at \( x=0 \), now shifts to \( x=2 \). The horizontal asymptote remains unchanged at \( y=0 \).
04

Sketch the Transformed Graph

With the vertical asymptote at \( x = 2 \) and the horizontal asymptote at \( y = 0 \), draw the branches of the hyperbola. The graph will resemble the original hyperbola, but shifted to the right by 2 units.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reciprocal Function
A reciprocal function is a specific type of rational function often expressed as \( y = \frac{1}{x} \). This function forms the basis for many transformations in graphing rational functions. It is defined wherever \( x eq 0 \) since division by zero is undefined. The graph of this function is a hyperbola with two asymptotes: a vertical asymptote at \( x = 0 \) and a horizontal asymptote at \( y = 0 \). Asymptotes are lines that the graph approaches but never actually touches.Bullets:
  • Key feature: the hyperbola approaches the asymptotes but never reaches them.
  • Symmetric about the origin, meaning it reflects equally in the opposite quadrants.
The reciprocal function helps in understanding more complex rational functions as transformations are typically applied to this base graph.
Horizontal Shift
A horizontal shift changes the position of a graph along the x-axis. For the function \( f(x) = \frac{1}{x-2} \), the parent function \( y = \frac{1}{x} \) experiences a shift. The value \( x-2 \) tells us that the graph has moved 2 units to the right. This is because we replace \( x \) with \( x - 2 \), effectively adjusting the input by shifting it.When graphing, remember:
  • A positive shift means moving to the right.
  • The vertical asymptote follows the shift, from \( x = 0 \) to \( x = 2 \) in this example.
Shifts like these are common and an easy way to adjust the graph's position without changing its overall shape.
Vertical Asymptote
A vertical asymptote is a vertical line that the graph of a function approaches but never touches or crosses. It usually indicates where a function is undefined. In our example, the transformation of the reciprocal function \( y = \frac{1}{x} \) to \( f(x) = \frac{1}{x-2} \) causes the vertical asymptote to shift from its position at \( x = 0 \) to \( x = 2 \).Things to remember about vertical asymptotes:
  • A vertical asymptote typically corresponds to values that make the denominator zero.
  • The behavior of the graph near an asymptote is a crucial feature in sketching rational functions.
Understanding asymptotes helps predict the graph's behavior without plotting numerous points.
Hyperbola Graph
The graph of a reciprocal function, such as \( y = \frac{1}{x} \), is a hyperbola. Hyperbolas are curves with two separate branches, each approaching asymptotes but not intersecting them. Transformations such as horizontal shifts adjust the location of these branches.Features of a Hyperbola:
  • The two branches occupy opposite quadrants, reflecting symmetry around the origin in the parent function.
  • In \( f(x) = \frac{1}{x-2} \), the branches are shifted right to align with the new vertical asymptote at \( x = 2 \), while maintaining the same shape and angles.
Visualizing these changes aids in understanding how transformations affect the fundamental appearance of the graph.

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Most popular questions from this chapter

For the given rational function \(f\) : \(\bullet\)Find the domain of \(f\). \(\bullet\)Identify any vertical asymptotes of the graph of \(y=f(x)\) \(\bullet\)Identify any holes in the graph. \(\bullet\)Find the horizontal asymptote, if it exists. \(\bullet\)Find the slant asymptote, if it exists. \(\bullet\)Graph the function using a graphing utility and describe the behavior near the asymptotes. $$f(x)=\frac{x^{3}-4 x^{2}-4 x-5}{x^{2}+x+1}$$

Translate the following into mathematical equations. At a constant pressure, the temperature \(T\) of an ideal gas is directly proportional to its volume \(V\).

Solve the rational inequality. Express your answer using interval notation. $$\frac{-x^{3}+4 x}{x^{2}-9} \geq 4 x$$

Solve the rational inequality. Express your answer using interval notation. $$\frac{3 x-1}{x^{2}+1} \leq 1$$

Graph the rational function by applying transformations to the graph of \(y=\frac{1}{x}\). $$h(x)=\frac{-2 x+1}{x}$$
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