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Chapter 1: Problem 2

For Activities 1 through \(6, \quad\) for each linear model a. give the slope of the line defined by the equation. b. write the rate of change of the function in a sentence of interpretation. c. evaluate and give a sentence of interpretation for \(f(0)\). The number of people of age \(x\) years in a certain country is \(f(x)=-0.5 x+3.2\) million people.

Short Answer

Expert verified
Slope: -0.5; Population decreases by 0.5 million/year; 3.2 million newborns.

Step by step solution

01

Identify the Slope

The linear function is given as \( f(x) = -0.5x + 3.2 \). In a linear equation of the form \( f(x) = mx + b \), the slope is represented by \( m \). Here, the slope \( m \) is \(-0.5\).
02

Interpret the Rate of Change

The rate of change in a linear function corresponds to the slope. Since the slope is \(-0.5\), it means that for each year increase in age, the population decreases by 0.5 million people.
03

Evaluate \(f(0)\)

Substitute \( x = 0 \) into the function: \( f(0) = -0.5(0) + 3.2 \). Simplifying this, \( f(0) = 3.2 \).
04

Interpret \(f(0)\)

\( f(0) \) represents the population of people who are 0 years old. The interpretation is that there are 3.2 million newborns in the country.

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Key Concepts

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

Slope of a Line
In the world of linear models, the term "slope" is key to understanding how variables interact. Let's consider the function given in the exercise:
\( f(x) = -0.5x + 3.2 \).
This equation is linear, and generally, such equations are expressed in the form \( f(x) = mx + b \), where \( m \) is the slope.
  • The slope essentially tells us how steep the line is and its direction.
  • For our function, the slope \(-0.5\) indicates a downward trend.
  • A negative slope implies that as one variable increases, the other decreases.
This means, for each additional year in age \( x \), the population decreases by 0.5 million people. So, the slope not only tells us how much change to expect, but also the direction of that change, shaping our understanding of how two variables are related.
Rate of Change
In linear models, the rate of change is synonymously represented by the slope. It measures how quickly changes occur between two variables. Let's delve deeper into what this means:
For the given function \( f(x) = -0.5x + 3.2 \), the slope is \(-0.5\).
  • This tells us that for every one unit increase in \( x \) (each additional year), the function value decreases by 0.5.
  • In the context of the problem, it means the number of people decreases by 0.5 million each year.
  • The rate of change tells us about the relationship dynamics between age and population.
Expressing this in simpler terms, if the age increases steadily, we expect the population to learn more about decreasing influence over time. The rate of change acts as a summary of the interaction between these two factors.
Function Evaluation
Function evaluation involves plugging a specific value into the function to find the output. In this problem, you are tasked with evaluating \( f(0) \).
This requires you to substitute \( x = 0 \) into the function provided:
\( f(0) = -0.5(0) + 3.2 \).
  • By simplifying this, we arrive at \( f(0) = 3.2 \).
  • This result tells us the number of people aged 0 in the country.
  • It's simply translating a mathematical expression into a real-world context.
For this problem, interpreting \( f(0) \), we find that 3.2 million represents the "newborn" population, showcasing how equations translate into meaningful real-world insights. Evaluating functions can thus allow us a snapshot into specific data points within a larger model.

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Most popular questions from this chapter

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