Problem 26 - College Algebra

Question

The bar graph shows that as costs changed over the decades, Americans devoted less of their budget to groceries and more to health care. (Graph cant copy) Find a linear function in slope-intercept form that models the given description. Each function should model the percentage of total spending, \(p(x),\) by Americans \(x\) years after 1950. In 1950 , Americans spent \(3 \%\) of their budget on health care. This has increased at an average rate of approximately \(0.22 \%\) per year since then.

Step-by-Step Solution

Answer

The linear function that models the given scenario in slope-intercept form is \(p(x) = 0.22x + 3\).

Step 1 Identify the y-intercept

Looking at the problem, it is stated that in 1950, the Americans used 3% of their budget on health care. In terms of the function \(p(x)\), this is the value of \(p(x)\) at \(x = 0\). Therefore, the y-intercept of the function is 3. This is written as \(b = 3\).

Step 2 Identify the slope

It is also stated in the problem that the spending on health care has increased at an average rate of approximately 0.22% per year. This value represents the rate of change, or the slope (m) of the function. This is written as \(m = 0.22\).

Step 3 Formulate the linear function

The general form of a linear function in slope-intercept form is \(y = mx + b\). Substituting the values for m and b that were identified from the given information, the function modeling the percentage of total spending on health care by Americans x years after 1950 is \(p(x) = 0.22x + 3\). Using this function, you can predict the percentage of total spending on health care by Americans in any given year after 1950 by substituting the number of years after 1950 for x.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is an efficient way to express a line on a graph. It is written as:

• \(y = mx + b\)

In this equation, \(m\) represents the slope of the line, showing how steep it is, while \(b\) is the y-intercept, indicating where the line crosses the y-axis.By using the slope-intercept form, you can quickly determine the behavior of the line and easily graph it.
In practice, it helps estimate changes over time, predict future values, and understand relationships between variables.

Rate of Change

The rate of change is the amount a variable changes over an interval of time. In the context of linear functions, this is represented by the slope, \(m\), in the equation of the line.

• In our exercise, the rate of change is 0.22%, reflecting the annual increase in budget for health care since 1950.

The rate of change is crucial because it indicates how one quantity changes in relation to another. It provides insights into trends and can be used to make future predictions.
When interpreting problems using linear functions, identifying the rate of change helps understand the dynamics at play.

Y-Intercept

The y-intercept is the point on a graph where the line crosses the y-axis. In the slope-intercept form \(y = mx + b\), it is represented by \(b\).

• For our problem, the y-intercept is given as 3, meaning in 1950 (\(x = 0\)), Americans spent 3% of their budget on health care.

Understanding the y-intercept is important in defining the starting point of a linear model in real-world situations. It helps signify the initial condition before any changes occur over time.
The y-intercept provides context to data by marking the baseline from which other values increase or decrease.

Budget Allocation

Budget allocation refers to how resources are distributed across different categories. In this exercise, we examined how Americans spent more on health care over the years, indicating a change in budget priorities.

• This change is represented by the linear function \(p(x) = 0.22x + 3\), giving insight into evolving spending habits.

Understanding how budget allocation shifts can help policymakers and individuals make informed economic decisions.
By modeling budget allocation with linear functions, it becomes easier to analyze and visualize these changes and make predictions about future trends.