/*! 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 3 What type of transformation of t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 3

What type of transformation of the graph of \(f(x)=5^{x}\) is the graph of \(f(x+1) ?\)

Short Answer

Expert verified
The graph of \(f(x+1)\) is a horizontal shift or translation of the graph of \(f(x) = 5^x\) one unit to the left.

Step by step solution

01

Recognize the Transformation

In the function \(f(x+1)\), the transformation is happening to the input or the 'x' value. That is, we are adding 1 to every x-value that we input into our function.
02

Interpret the Transformation

In general, if we replace \(x\) with \(x+c\) in a function, the graph of the function shifts c units to the left. So for the function \(f(x+1)\), this means the graph of \(f(x) = 5^x\) will shift one unit to the left.
03

Confirm the Transformation

Let's confirm this by testing a value. Take \(x=1\) for example, then \(f(1) = 5^1 = 5\), whereas \(f(1+1) = f(2) = 5^2 = 25\). When \(x\) was increased by 1, the resulting value became that of where \(x=2\) in the original function, effectively shifting the graph one unit to the left.

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.

Graph Shifts
When discussing the transformation of functions, one fundamental and commonly encountered modification is the horizontal shift of a graph. A graph shift occurs when every point of a function moves in the same direction along the x or y-axis. In the context of our function,
  • The transformation of the graph of the function \( f(x) = 5^x \) to \( f(x+1) \) involves a horizontal shift.
  • This particular shift is to the left.
This is because we add 1 to the input variable, \( x \), turning it into \( x + 1 \). A general rule to remember with horizontal shifts is:
  • If you replace \( x \) with \( x + c \) in a function, the graph shifts \( c \) units to the left.
  • If you replace \( x \) with \( x - c \), it shifts \( c \) units to the right.
Visualizing this can be particularly useful in understanding different transformations and effectively graphing functions.
Exponential Functions
Exponential functions are a crucial part of mathematics with profound implications in science, finance, and many other areas. An exponential function can be written in the form of \( f(x) = a^x \), where:
  • \( a \) is a constant base greater than 0 and not equal to 1.
  • \( x \) is the exponent, which is the variable that determines how many times the base is multiplied by itself.
The base determines the growth or decay rate:
  • If \( a > 1 \), the function exhibits exponential growth.
  • If \( 0 < a < 1 \), it shows exponential decay.
For our specific function, \( f(x) = 5^x \), with a base of 5, the function demonstrates rapid growth as \( x \) increases.Understanding these characteristics of exponential functions is essential when analyzing their graph and predicting behavior over time.
Function Transformation
Function transformation involves changing the appearance or position of a graph by altering its equation in specific ways.For instance, in our exercise with the function \( f(x) = 5^x \), transforming to \( f(x+1) \) changes how the function behaves:
  • This change results from modifying the function's input by adding 1, effectively shifting the graph horizontally.
  • The transformation doesn't change the scale or shape but influences the graph's position.
Well-known types of function transformations include:
  • Shifting: Moving the graph horizontally or vertically.
  • Stretching or Compressing: Changing the function's rate of increase or decrease.
  • Reflecting: Flipping the graph across axes.
By mastering these transformations, one can easily tackle complex functions and sketch their graphs more effectively, a skill useful across various mathematical contexts.

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

The numbers \(y\) of commercial banks in the United States from 2007 through 2013 can be modeled by $$y=11,912-2340.1 \ln t, \quad 7 \leq t \leq 13$$ where \(t\) represents the year, with \(t=7\) corresponding to \(2007 .\) In what year were there about 6300 commercial banks? (Source: Federal Deposit Insurance Corp.)

Use the regression feature of a graphing utility to find a logarithmic model \(y=a+b \ln x\) for the data and identify the coefficient of determination. Use the graphing utility to plot the data and graph the model in the same viewing window. $$(3,14.6),(6,11.0),(9,9.0),(12,7.6),(15,6.5)$$

Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$\frac{1+\ln x}{2}=0$$

Use a graphing utility to create a scatter plot of the data. Decide whether the data could best be modeled by a linear model, an exponential model, or a logarithmic model. $$(1,4.4),(1.5,4.7),(2,5.5),(4,9.9),(6,18.1),(8,33.0)$$

Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$-x e^{-x}+e^{-x}=0$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

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.