Problem 49 - College Algebra with Modeling and Visualization

Question

Exercises \(49-54:\) Write a formula for a linear function that models the situation. Choose both an appropriate name and an appropriate variable for the function. State what the input variable represents and the domain of the function. Assume that the domain is an interval of the real numbers. U.S. Homes with Internet In 2006 about \(68 \%\) of U.S. homes had Internet access. This percentage was expected to increase, on average, by 1.5 percentage points per year for the next 4 years. (Source: 2007 Digital Future Report)

Step-by-Step Solution

Answer

I(t) = 68 + 1.5t; Domain: [0, 4].

Step 1: Define Variables and Context

Let's determine the input variable and what it represents. Since we are dealing with years after 2006 and internet access percentages, let \( t \) be the number of years since 2006. The variable \( I(t) \) will represent the percentage of U.S. homes with internet access \( t \) years after 2006.

Step 2: Identify the Initial Value

Identify the given information about the initial condition. In this case, in 2006, which is when \( t = 0 \), 68% of U.S. homes had internet access. This represents the initial value, \( I(0) = 68 \).

Step 3: Determine the Rate of Change

The percentage of homes with internet access is expected to increase by 1.5 percentage points per year. This means that the rate of change, or slope, is 1.5. In the linear function, this is the coefficient of \( t \).

Step 4: Write the Linear Function

Using the initial value and the rate of change, the linear function for the model is expressed as:\[I(t) = 68 + 1.5t\]This equation models the percentage of homes with internet access \( t \) years after 2006.

Step 5: State the Domain

Since the internet access percentage is expected to increase over the next 4 years, the domain of the function is the interval \( [0, 4] \), meaning \( t \) can be any real number between 0 and 4, inclusive.

Key Concepts

Rate of Change

In the context of linear functions, the 'rate of change' is a crucial concept. It tells us how much one variable changes in relation to another. Here, the rate of change refers to the increase in the percentage of U.S. homes with Internet access each year. In the given problem, it's mentioned that this rate of change is 1.5 percentage points per year. This means, each year, the number of homes with internet access rises by an additional 1.5% compared to the previous year.

• The rate of change is also known as the 'slope' or 'gradient' in mathematics.
• In a function, it's the coefficient of the variable.
• It provides a measure of the steepness and direction of the line on a graph.

This value is represented by 'm' in the typical linear equation form, which is: \[ y = mx + b \] Here, 1.5 is similar to 'm', indicating the linear growth rate.

Initial Value

Every linear function has an 'initial value,' which is where the line starts on the y-axis. It's like a starting point or baseline before any changes occur. For the exercise, the initial value is the percentage of homes with Internet access in 2006, which is 68%.

• This value occurs when the input variable is zero, i.e., at the start of the analysis.
• In this linear function, it is denoted as \( I(0) \).
• The initial value can also be considered the 'y-intercept' of the function.

This figure provides a baseline that gives context to the change over time. Without knowing where we start from, it's hard to gauge the impact of the growth or decrease.

Domain of a Function

The 'domain of a function' outlines all the possible input values (usually represented by 'x' or another variable) that can be applied within a problem. It helps define the scope of the function and gives a limit to the values that we can substitute into the function.

• Here, the domain is stated as the interval [0, 4].
• This interval indicates that the function is applicable for any real number between 0 and 4, inclusive.
• This means you can analyze changes from the starting year (2006) up until four years after (2010).

The domain ensures that the function remains relevant, preventing unrealistic predictions beyond the provided data set.

Internet Access Growth

The ongoing increase in the percentage of homes with Internet access showcases a trend in 'Internet Access Growth'. It's essential to model these trends to understand long-term developments and adapt to changes in technological accessibility.

• Growth represents the overall increase in data or values over time.
• The model follows a straight-line pattern due to the consistency in growth (1.5% each year).
• It reflects broader societal shifts towards higher connectivity.

In this linear model, the function \( I(t) = 68 + 1.5t \) captures the growth trend effectively by illustrating how many more homes acquire Internet access annually.