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Chapter 1: Problem 1

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 cost to rent a newly released movie is \(f(x)=0.3 x+5\) dollars, where \(x\) is the number of years since 2010 .

Short Answer

Expert verified
a. Slope: 0.3 b. The cost increases by $0.3 each year after 2010. c. In 2010, the rental cost was $5.

Step by step solution

01

Identify the Slope

The linear model given is \(f(x) = 0.3x + 5\). In this equation, the slope is the coefficient of \(x\), which is \(0.3\). The slope represents the rate at which the cost to rent a movie increases per year after 2010.
02

Interpret the Rate of Change

The slope of \(0.3\) means that for every year after 2010, the cost to rent a newly released movie increases by \(0.3\) dollars.
03

Evaluate \(f(0)\)

To find \(f(0)\), substitute \(x = 0\) into the equation \(f(x) = 0.3x + 5\). This gives \(f(0) = 0.3(0) + 5 = 5\).
04

Interpret \(f(0)\)

The value \(f(0) = 5\) indicates that the cost to rent a newly released movie in the year 2010 was 5 dollars.

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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 context of linear functions, the slope of a line is a critical element that determines how steep the line is. It is essentially the "rate of change" of the function. When you look at a linear equation such as \(f(x) = 0.3x + 5\), the number that multiplies the \(x\) term (in this case, \(0.3\)) is the slope.
The slope tells us how much the output value (what \(f(x)\) equals) changes as the input value \(x\) changes.
In our example with the movie rental costs, the slope of \(0.3\) means that every year, the cost increases by \(0.3\) dollars.
  • A positive slope (as in \(0.3\)) means the line rises as it moves from left to right.
  • A larger number for the slope would mean a steeper increase.
  • If the slope were negative, the line would fall, indicating a decrease in value.
This concept of slope is vital in evaluating trends, showing how rapid or gradual changes occur in different scenarios.
Rate of Change
Rate of change is a fundamental concept in mathematics that describes how one quantity changes in relation to another. In linear functions, like our example \(f(x) = 0.3x + 5\), this rate is constant and indicated by the slope, which is \(0.3\) here.
The rate of change helps us understand how quick or slow a change is happening over time.
For the rental cost function:
  • The rate of change is \(0.3\) dollars per year.
  • Tells us that starting in 2010, the cost to rent a movie will increase by \(0.3\) dollars each following year.
This rate is essential in predicting future values, helping individuals and businesses make informed decisions based on expected changes.
Understanding rate of change allows you to describe real-world relationships accurately and manipulate the function to project various outcomes.
Function Evaluation
Function evaluation involves substituting values into a function to get specific results. It helps determine the output for a given input, providing crucial insights into the behavior of mathematical models.
Let's revisit our linear function, \(f(x) = 0.3x + 5\). The task of evaluating \(f(0)\) requires inputting \(x = 0\) into the equation, giving:
\[f(0) = 0.3(0) + 5 = 5\]
This calculation shows that at \(x = 0\), which represents the year 2010, the cost of a new movie rental was \(5\) dollars.
  • Function evaluation is used to find specific values that are critical for understanding the function's real-world implications.
  • By evaluating at different points, you can create a table of values or graph to visualize the behavior of a function.
Understanding how to evaluate functions provides a practical way to process information and make predictions based on a mathematical model.

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Most popular questions from this chapter

Sleep Time (Women) Until women reach their mid-60s, they tend to get less sleep per night as they age. The average number of hours (in excess of eight hours) that a woman sleeps per night can be modeled as $$ s(w)=2.697\left(0.957^{w}\right) \text { hours } $$ when the woman is \(w\) years of age, \(15 \leq w \leq 64\). (Source: Based on data from the Bureau of Labor Statistics) a. How much sleep do women of the following ages get: \(15,20,40,64 ?\) b. Using the results from part \(a\), write a model giving age as a function of input \(s\) where \(s+8\) hours is the average sleep time.

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