/*! 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 2 Create a linear or exponential f... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 2

Create a linear or exponential function based on the given conditions. a. A function with an average rate of change of 3 and a vertical intercept of 4 b. A function with growth factor of 3 and vertical intercept of \(4 .\) c. A function with slope of \(4 / 3\) and initial value of 5 . d. A function with initial value of 5 and growth factor of \(4 / 3\).

Short Answer

Expert verified
a. \(f(x) = 3x + 4\); b. \(f(x) = 4 \times 3^x\); c. \(f(x) = \frac{4}{3}x + 5\); d. \(f(x) = 5 \times \left(\frac{4}{3}\right)^x\).

Step by step solution

01

Identify the type of function

Determine whether a linear or exponential function is needed based on the average rate of change or growth factor.
02

Step 2a: Create a linear function with average rate of change of 3 and vertical intercept of 4

The formula for a linear function is given by \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Here, \(m = 3\) and \(b = 4\). So the function is: \(f(x) = 3x + 4\)
03

Step 2b: Create an exponential function with growth factor of 3 and vertical intercept of 4

The formula for an exponential function is given by \(f(x) = ab^x\), where \(a\) is the initial value and \(b\) is the growth factor. Here, \(a = 4\) and \(b = 3\). So the function is: \(f(x) = 4 \times 3^x\)
04

Step 2c: Create a linear function with slope of \(\frac{4}{3}\) and initial value of 5

The formula for a linear function is \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Here \(m = \frac{4}{3}\) and \(b = 5\). So the function is: \(f(x) = \frac{4}{3}x + 5\)
05

Step 2d: Create an exponential function with initial value of 5 and growth factor of \(\frac{4}{3}\)

The formula for an exponential function is \(f(x) = ab^x\), where \(a\) is the initial value and \(b\) is the growth factor. Here, \(a = 5\) and \(b = \frac{4}{3}\). So the function is: \(f(x) = 5 \times \left(\frac{4}{3}\right)^x\)

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.

average rate of change
In mathematics, the average rate of change of a function measures how much the function's value changes on average over a specified interval. For a linear function, this is simply the slope.
The formula for the average rate of change between two points \(x_1\) and \(x_2\) is \[ \frac{f(x_2) - f(x_1)}{x_2 - x_1} \]
In the exercise, we needed a linear function with an average rate of change (or slope) of 3 and a vertical intercept of 4. The function thus translates to \(f(x) = 3x + 4\).
The average rate of change informs us that for every one unit increase in x, the value of f(x) increases by 3 units. For example:
  • If x increases from 0 to 1, f(x) increases from 4 to 7.
  • If x increases from 1 to 2, f(x) increases from 7 to 10.
growth factor
Growth factor is key for exponential functions and is distinct from the notion of slope in linear functions. It represents the constant ratio by which a quantity multiplies over equal intervals.
For an exponential function \(f(x) = ab^x\), \(b\) stands for the growth factor.
In the problem, we had an exponential function where the growth factor is 3 and the vertical intercept (initial value) is 4. This results in the function \(f(x) = 4 \times 3^x\).
Each increase of one unit in x results in the value of the function being multiplied by the growth factor. For example:
  • When x = 1, f(x) is \(4 \times 3^1 = 12\).
  • When x = 2, f(x) becomes \(4 \times 3^2 = 36\).
initial value
The initial value is the value of the function when the input (x) is zero. It represents where the line or curve intersects the y-axis.
For linear functions of the form \(f(x) = mx + b\), the initial value is equal to \(b\), the vertical intercept, while for exponential functions of the form \(f(x) = ab^x\), the initial value is \(a\).
Let's examine the exercise:
  • In the linear function \(f(x) = \frac{4}{3} x + 5\), the initial value is 5.
  • In the exponential function, \(f(x) = 5 \times \frac{4}{3}^x\), the initial value is also 5.

These values are crucial as they provide information about the starting point of the function.
slope
The slope of a linear function portrays the rate at which the value of the function is changing. In mathematical terms, it is the coefficient of x in the linear function \(f(x) = mx + b\).
The slope formula is \ m = \frac{change \, in \, y}{change \, in \, x} \
From the exercise, we needed a linear function with a slope of \ \frac{4}{3} \ and an initial value of 5, resulting in \(f(x) = \frac{4}{3} x + 5\)
Here, the slope \ \frac{4}{3} \ indicates that for every increase of 1 unit in x, the function value (y) increases by \ \frac{4}{3} \ units.
vertical intercept
The vertical intercept, also known as the y-intercept, is pivotal in both types of functions. It indicates where the graph intersects the y-axis.
For both linear and exponential functions, this occurs when x = 0.
  • In the linear formula \(f(x) = mx + b\), the intercept is \(b\).
  • In the exponential formula \(f(x) = ab^x\), the intercept is \(a\).

Review the exercise's functions:
  • Linear function \(f(x) = 3x + 4\) has intercept 4.
  • Exponential function \(f(x) = 4 \times 3^x\) begins at 4.

This value represents where each function 鈥榮tarts鈥� on the vertical axis.

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

A pollutant was dumped into a lake, and each year its amount in the lake is reduced by \(25 \%\). a. Construct a general formula to describe the amount of pollutant after \(n\) years if the original amount is \(A_{0}\). b. How long will it take before the pollution is reduced to below \(1 \%\) of its original level? Justify your answer.

China is the most populous country in the world. In 2000 it had about 1.262 billion people. By 2005 the population had grown to 1.306 billion. Use this information to construct models predicting the size of China's population in the future. a. Identify your variables and units. b. Construct a linear model. c. Construct an exponential model. d. What will China's population be in 2050 according to each of your models?

(Requires technology to find a best-fit function.) Reliable data on Internet use are hard to find, but World Telecommunications Indicators cites estimates of 3 million U.S. users in 1991,30 million in 1996,166 million in 2002,199 million in 2004 and 232 million in 2007 . a. Use technology to plot the data, and generate a best-fit linear and a best- fit exponential function for the data. Which do you think is the better model? b. What would the linear model predict for Internet usage in \(2010 ?\) What would the exponential model predict? c. Internet use: Go online and see if you can find the number of current internet users in the U.S. Which of your models turned out to be more accurate?

The average female adult Ixodes scapularis (a deer tick that can carry Lyme disease) lives only a year but can lay up to 10,000 eggs right before she dies. Assume that the ticks all live to adulthood, and half are females that reproduce at the same rate and half the eggs are male. a. Describe a formula that will tell you how many female ticks there will be in \(n\) years, if you start with one impregnated tick. b. How many male ticks will there be in \(n\) years? How many total ticks in \(n\) years? c. If the surface of Earth is approximately \(5.089 \cdot 10^{14}\) square meters and an adult tick takes up 0.5 square centimeters of land, approximately how long would it take before the total number of ticks stemming from one generation of ticks would cover the surface of Earth?

Each of two towns had a population of 12,000 in \(1990 .\) By 2000 the population of town A had increased by \(12 \%\) while the population of town B had decreased by \(12 \%\). Assume these growth and decay rates continued. a. Write two exponential population models \(A(T)\) and \(B(T)\) for towns A and \(\mathrm{B}\), respectively, where \(T\) is the number of decades since 1990 . b. Write two new exponential models \(a(t)\) and \(b(t)\) for towns A and \(\mathrm{B}\), where \(t\) is the number of years since 1990 . c. Now find \(A(2), B(2), a(20)\), and \(b(20)\) and explain what you have found.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

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.