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Chapter 3: Problem 56

Graph the functions by using transformations of the graphs of \(y=\frac{1}{x}\) and \(y=\frac{1}{x^{2}}\). $$ g(x)=\frac{1}{x+4} $$

Short Answer

Expert verified
Shift the graph of \(y=\frac{1}{x}\) 4 units to the left.

Step by step solution

01

Understand the Parent Function

The parent function is given as \(y=\frac{1}{x}\). Its graph is a hyperbola with two branches, asymptotes at \(x=0\) and \(y=0\), and it is symmetric about the origin.
02

Identify the Transformation

For \(g(x)=\frac{1}{x+4}\), the transformation involves a horizontal shift. Here, the graph shifts left by 4 units due to the \(+4\) inside the function.
03

Apply the Horizontal Shift

To graph \(g(x)=\frac{1}{x+4}\), take the parent function \(y=\frac{1}{x}\) and move every point 4 units to the left. The vertical asymptote shifts from \(x=0\) to \(x=-4\).
04

Draw the Transformed Graph

Plot the hyperbola with the new position. The graph will have vertical asymptote at \(x=-4\) and a horizontal asymptote still at \(y=0\). Ensure the branches extend towards the asymptotes without touching them.

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Key Concepts

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

parent function
The parent function in this exercise is denoted as \( y = \frac{1}{x} \). This particular function is a basic form known as a rational function. Its graph represents a hyperbola, a type of curve with two distinct branches.

The hyperbola is symmetric about the origin, meaning any point \((x, y)\) on the curve will have a corresponding point \((-x, -y)\).
As for its significant features:
  • The vertical asymptote is at \( x = 0 \).
  • The horizontal asymptote is at \( y = 0 \).
The asymptotes are lines that the graph approaches but never touches. This behavior helps us understand how the function behaves at extreme values of \(x\).
horizontal shift
Transforming functions often involves shifting their graphs. In the exercise, we're dealing with a horizontal shift. The given function is \( g(x)=\frac{1}{x+4} \).
When analyzing \( x+4 \) inside the function, we infer that it shifts the parent function \( y = \frac{1}{x} \) horizontally. Here, adding \( 4 \) means moving the graph to the left by 4 units.
This transformation modifies our vertical asymptote from \( x=0 \) to \( x=-4 \), while the horizontal asymptote remains unchanged at \( y=0 \). When performing the shift:
  • Move every point on the graph 4 units left.
  • Ensure the shape of the hyperbola remains consistent.
hyperbola
A hyperbola is a unique curve in mathematics characterized by two separate branches. The parent function \( y = \frac{1}{x} \) produces a hyperbola.

Understanding its structure:
  • It has two symmetric branches.
  • The branches approach the asymptotes but never intersect them.
  • The hyperbola is steep near the origin and flattens out as \( x \) moves farther from zero.
With transformations, you can shift and reshape hyperbolas, but their fundamental properties remain unchanged.
In \( g(x)=\frac{1}{x+4} \), the original hyperbola is shifted left by 4 units, maintaining its asymptotic behavior relative to the new vertical asymptote at \( x=-4 \).
asymptotes
Asymptotes are lines that a graph approaches but never actually touches. In the given parent function \( y = \frac{1}{x} \), there are two important asymptotes to consider:
  • Vertical asymptote at \( x=0 \).
  • Horizontal asymptote at \( y=0 \).
These lines help define the shape and boundaries of the hyperbola.
When a horizontal shift occurs, such as with \( g(x) = \frac{1}{x+4} \), the vertical asymptote moves accordingly while the horizontal asymptote remains constant.
After shifting the function 4 units to the left, the new vertical asymptote will be at \( x = -4 \) instead of \( x = 0 \), but the horizontal asymptote stays at \( y=0 \).

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