/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 38 For the following exercises, use... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 38

For the following exercises, use the given transformation to graph the function. Note the vertical and horizontal asymptotes. The reciprocal squared function shifted down 2 units and right 1 unit.

Short Answer

Expert verified
Vertical asymptote at \( x = 1 \); horizontal asymptote at \( y = -2 \).

Step by step solution

01

Understand the Base Function

The base function for this transformation is the reciprocal squared function, which is given by \( f(x) = \frac{1}{x^2} \). Its graph includes vertical asymptotes at \( x = 0 \) and a horizontal asymptote at \( y = 0 \).
02

Apply the Horizontal Shift

The function is shifted right by 1 unit. To account for this transformation, replace \( x \) with \( x - 1 \). This results in the new function \( f(x) = \frac{1}{(x-1)^2} \). The vertical asymptote moves from \( x = 0 \) to \( x = 1 \).
03

Apply the Vertical Shift

The function is shifted down by 2 units. Subtract 2 from the previous step's result. The updated function is \( f(x) = \frac{1}{(x-1)^2} - 2 \). This affects the horizontal asymptote, moving it from \( y = 0 \) to \( y = -2 \).
04

Note the Asymptotes

After applying both transformations, the function \( f(x) = \frac{1}{(x-1)^2} - 2 \) has a vertical asymptote at \( x = 1 \) and a horizontal asymptote at \( y = -2 \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Reciprocal Squared Function
A reciprocal squared function is a unique mathematical function that takes the form \( f(x) = \frac{1}{x^2} \). This function is interesting because its values depend on the reciprocal of the square of \( x \). As \( x \) approaches zero, the denominator grows smaller, causing the function's value to increase sharply.
  • The graph of this function is always positive, since squaring any real number yields a non-negative result.
  • It features a U-shaped curve, symmetric about the y-axis due to having even powers of \( x \).
When transforming such a function, key characteristics like its asymptotes and general shape provide crucial guidance. Understand how each transformation properly alters the graph to predict the final look accurately.
Vertical Asymptote
Vertical asymptotes are lines where a function approaches infinity; they signal non-existence for real values at specific \( x \)-values. In the reciprocal squared function \( f(x) = \frac{1}{x^2} \), the vertical asymptote naturally occurs at \( x = 0 \), because dividing by zero is undefined.
  • When applying horizontal shifts, like a move right by 1 unit, the vertical asymptote shifts as well. So our modified function \( f(x) = \frac{1}{(x-1)^2} \) has a vertical asymptote at \( x = 1 \).
  • This transformation doesn't alter the infinite qualities of the curve near the asymptote but merely dictates the specific \( x \)-value not included in the domain.
Locating vertical asymptotes is essential in sketching rational function graphs since they reveal critical points of interest and behavior as \( x \) approaches those values.
Horizontal Asymptote
Horizontal asymptotes indicate the value a function tends to as \( x \) becomes very large or very small. In reciprocal squared functions like \( f(x) = \frac{1}{x^2} \), the horizontal asymptote is \( y = 0 \) because as \( x \) increases, \( \frac{1}{x^2} \) diminishes to zero.
  • After a vertical shift occurs, the entire function graph moves parallel to the y-axis. For the example shift down by 2 units in \( f(x) = \frac{1}{(x-1)^2} - 2 \), makes the horizontal asymptote appear at \( y = -2 \).
  • This shift doesn't change the shape of the curve, yet it lowers the asymtote's nobility as \( x \) elongates in magnitude.
Understanding the role of horizontal asymptotes in predicting long-term behavior clarifies how functions extend into the indefinite right and left directions on the Cartesian plane.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies jointly as \(x, z,\) and \(w\) and when \(x=1, \quad z=2, \quad w=5,\) then \(y=100\).

For the following exercises, use the given information to answer the questions. The current in a circuit varies inversely with its resistance measured in ohms. When the current in a circuit is 40 amperes, the resistance is 10 ohms. Find the current if the resistance is 12 ohms.

For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies inversely as \(x\) and whe \(x=4, \quad y=2\).

For the following exercises, use a graph to help determine the domain of the functions. $$ f(x)=\sqrt{\frac{(x+1)(x-1)}{x}} $$

For the following exercises, determine the function described and then use it to answer the question. The surface area, \(A,\) of a cylinder in terms of its radius, \(r,\) and height, \(h,\) is given by \(A=2 \pi r^{2}+2 \pi r h .\) If the height of the cylinder is 4 feet, express the radius as a function of \(A\) and find the radius if the surface area is 200 square feet.
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.