/*! 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 54 Write a formula for the function... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 54

Write a formula for the function that results when the given toolkit function is transformed as described. \(f(x)=\frac{1}{x}\) vertically stretched by a factor of \(8,\) then shifted to the right 4 units and up 2 units.

Short Answer

Expert verified
The function is \( f(x) = \frac{8}{x-4} + 2 \).

Step by step solution

01

Identify the base function

The given function is a reciprocal function, which is expressed as \( f(x) = \frac{1}{x} \). Our task is to apply certain transformations to this function.
02

Apply the vertical stretch

To vertically stretch the function by a factor of 8, we multiply the entire function by 8. So the transformed function becomes \( 8 \cdot f(x) = 8 \cdot \frac{1}{x} = \frac{8}{x} \).
03

Shift the graph to the right

To shift the function 4 units to the right, we replace \( x \) with \( x-4 \). Therefore, the function becomes \( \frac{8}{x-4} \).
04

Shift the graph upwards

To shift the function up 2 units, we add 2 to the entire function. The final transformed function is \( f(x) = \frac{8}{x-4} + 2 \).

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 Function
A reciprocal function is a type of rational function usually expressed in the form \( f(x) = \frac{1}{x} \). It is called "reciprocal" because it involves taking the reciprocal (or the multiplicative inverse) of the variable \( x \). This function has some unique characteristics:
  • It has two asymptotes, which are lines that the graph of the function approaches but never touches. For \( f(x) = \frac{1}{x} \), these are the x-axis (horizontal asymptote) and y-axis (vertical asymptote).
  • It will not cross the axes, as the value of \( f(x) \) becomes infinite as \( x \) approaches zero.
  • The graph of \( f(x) = \frac{1}{x} \) will be symmetrical along the line \( y = x \), forming a hyperbola.
Understanding this function is crucial as it forms the base for understanding transformed versions of the function, such as those involving stretches and shifts.
Vertical Stretch
A vertical stretch changes the steepness or height of a graph without altering its horizontal dimensions. To achieve a vertical stretch of a function \( f(x) \), you multiply the entire function by a positive constant \( a \) where \( a > 1 \). This multiplication increases the distance of the points from the x-axis, making the graph taller.
  • In the example \( f(x) = \frac{1}{x} \), a vertical stretch by a factor of 8 would transform it to \( g(x) = 8 \cdot f(x) = \frac{8}{x} \).
  • This transformation means each point's distance from the x-axis has increased eightfold, making the curve steeper around the origin.
  • Despite this change, the asymptotes remain unchanged, keeping their positions along the x-axis and y-axis.
Vertical stretches are fundamental in creating diverse graph shapes from a basic function, allowing for more dynamic interpretation of data.
Horizontal Shift
A horizontal shift moves the graph of a function left or right without altering its shape. To shift a function to the right by \( h \) units, substitute \( x \) with \( x-h \) in the function's formula. Conversely, substituting \( x \) with \( x+h \) will shift it to the left by \( h \) units.
  • For \( f(x) = \frac{1}{x} \), a right shift of 4 units modifies it to \( g(x) = \frac{8}{x-4} \).
  • This adjustment moves every point on the graph 4 units to the right, including its vertical asymptote, which shifts from \( x = 0 \) to \( x = 4 \).
  • While the horizontal asymptote remains unchanged at \( y = 0 \), the overall placement of the graph shifts horizontally.
Understanding horizontal shifts is important for accurately placing a graph according to specific requirements or data analysis.
Vertical Shift
A vertical shift involves moving the entire graph up or down along the y-axis. To shift a function vertically, you can add or subtract a constant from the whole function.
  • If you add \( k \) to \( f(x) \), the function shifts upward by \( k \) units; subtracting \( k \) shifts it downward.
  • For the function \( g(x) = \frac{8}{x-4} \), adding 2 results in the new function \( h(x) = \frac{8}{x-4} + 2 \).
  • This upward shift moves every point on the graph 2 units higher and raises the horizontal asymptote from \( y = 0 \) to \( y = 2 \).
Vertical shifts are essential for modifying how graphs align on a coordinate plane, especially in practical applications like fitting data to particular models.

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

Describe how each function is a transformation of the original function \(f(x)\) $$ f(x)-2 $$

Select all of the following tables which represent \(y\) as a function of \(x\). a. $$ \begin{array}{|l|l|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline 0 & -2 \\ \hline 3 & 1 \\ \hline 4 & 6 \\ \hline 8 & 9 \\ \hline 3 & 1 \\ \hline \end{array} $$ b. $$ \begin{array}{|l|l|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline-1 & -4 \\ \hline 2 & 3 \\ \hline 5 & 4 \\ \hline 8 & 7 \\ \hline 12 & 11 \\ \hline \end{array} $$ c. $$ \begin{array}{|l|l|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline 0 & -5 \\ \hline 3 & 1 \\ \hline 3 & 4 \\ \hline 9 & 8 \\ \hline 16 & 13 \\ \hline \end{array} $$ d. $$ \begin{array}{|l|l|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline-1 & -4 \\ \hline 1 & 2 \\ \hline 4 & 2 \\ \hline 9 & 7 \\ \hline 12 & 13 \\ \hline \end{array} $$

Describe how each formula is a transformation of a toolkit function. Then sketch a graph of the transformation. $$ f(x)=4(x+1)^{2}-5 $$

For each table below, select whether the table represents a function that is increasing or decreasing, and whether the function is concave up or concave down. $$ \begin{array}{|l|l|} \hline \boldsymbol{x} & \boldsymbol{h}(\boldsymbol{x}) \\ \hline 1 & 300 \\ \hline 2 & 290 \\ \hline 3 & 270 \\ \hline 4 & 240 \\ \hline 5 & 200 \\ \hline \end{array} $$

The graph of \(f(x)=2^{x}\) is shown. Sketch a graph of each transformation of \(f(x)\) $$ g(x)=-2^{x}+1 $$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Applied 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.