Problem 28 - Precalculus Essentials

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. 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 A mericans \(x\) years after \(1950 .\) (GRAPH CAN'T COPY) 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 percentage of total spending on health care by Americans x years after 1950 is p(x) = 0.22x + 3

Step 1: Identify the slope

The rate of change, also known as the slope, indicates the amount that the percentage of total spending for health care increases each year. In this case, the increase is 0.22% per year, so the slope (m) is 0.22.

Step 2: Identify the y-intercept

The y-intercept is the value of the target variable when x equals 0. In this case, in the year 1950 (x=0), the percentage of total spending on health care was 3%. Thus, the y-intercept (b) is 3.

Step 3: Write the linear function

Once we have the slope and the y-intercept, we can plug these values into the slope-intercept form to define the linear function. The linear function becomes p(x) = 0.22x + 3 where p(x) is the percentage of total spending on health care x years after 1950.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to represent linear equations. It is one of the most common forms used in math because it clearly shows both the slope and the y-intercept of a line. This form is written as \( y = mx + b \), where:

• \( y \) represents the dependent variable;
• \( m \) is the slope of the line;
• \( x \) is the independent variable, often representing time or another factor;
• \( b \) is the y-intercept, the value of \( y \) when \( x \) is 0.

This form is highly useful because it gives us two pieces of key information about the line, directly in the equation. Understanding how to break it down into its components helps in graphing lines and understanding linear changes. In our specific example, the slope-intercept form helps model the change in American spending on healthcare over time. Using \( p(x) = 0.22x + 3 \) lets us see how this spending evolves from the base year, 1950.

Slope

The slope of a line in a linear function is crucial because it represents the rate of change. In the slope-intercept form \( y = mx + b \), the \( m \) is the slope. It tells us how much \( y \) changes for a unit change in \( x \). Think of the slope as the steepness of a hill; a larger slope means a steeper hill.
In our example with healthcare spending, the slope is \( 0.22 \). This means that for each year after 1950, the percentage of spending on healthcare increases by 0.22%.
The slope can be positive or negative. A positive slope, like in our case, indicates an upward trend, implying increased spending over time. Understanding this concept helps to predict future values and understand the pattern changes over time.

Y-Intercept

The y-intercept is another fundamental concept in linear functions. Represented as \( b \) in the slope-intercept form \( y = mx + b \), it signifies the point where the line crosses the y-axis. This is the starting value of the dependent variable when \( x \) is zero.
In the context of modeling budget spending, the y-intercept shows the initial percentage of total budget spending in the base year. For our healthcare example, \( b = 3 \), which means that at the year 1950, healthcare expenses were 3% of the total budget.

• The y-intercept is essential as it often represents initial conditions or starting points in real-world situations.
• It provides an understanding of the base level before any changes made by the slope take effect.

Understanding the y-intercept allows you to get a complete picture of the situation you're modeling, giving insights into how things started before changes began.