Problem 61 - Precalculus: Functions and Graphs

Question

Medicare recipients The function \(f\) given by $$ f(x)=-0.000015 z^{3}-0.005 z^{2}+0.75 z+23.5 $$ where \(z=x-1973,\) approximates the total number of Medicare recipients in millions, from \(x=1973\) to \(x=2005 .\) There were \(23,545,363\) Medicare recipients in 1973 and \(42,394,926\) in 2005 (a) Graph \(f,\) and discuss how the number of Medicare recipients has changed over this time period. (b) Create a linear model similar to \(f\) that approximates the number of Medicare recipients. Which model is more realistic?

Step-by-Step Solution

Answer

The cubic model captures variations better; both models show increasing recipients.

Step 1: Understanding the Function and Its Domain

The function given is \( f(z) = -0.000015z^3 - 0.005z^2 + 0.75z + 23.5 \), where \( z = x - 1973 \). This signifies that \( x \) is the year, and \( z \) represents the years since 1973. The domain for \( x \) is from 1973 to 2005, translating to \( 0 \leq z \leq 32 \). We need to graph this function over this domain.

Step 2: Graphing the Function

To visualize \( f(z) \), plot \( f(z) \) on a graph with the x-axis representing the years since 1973 (\( z \)) and the y-axis representing the number of Medicare recipients (in millions). You should note the end points such as \( f(0) = 23.5 \) million and \( f(32) \approx 42.4 \) million, which align approximately with the given data points for 1973 and 2005.

Step 3: Analyzing the Graph

Observe the shape of the graph between \( z = 0 \) and \( z = 32 \). It should be a cubic curve starting from 23.5 million upwards to around 42.4 million by 2005, indicating an increase in Medicare recipients over the years. This upward trend reflects the increase in the number of eligible recipients due to factors such as the aging population.

Step 4: Creating a Linear Approximation

To create a linear model, use the two points provided: (1973, 23.545363) and (2005, 42.394926). Convert these to \( z \)-values: (0, 23.545363) and (32, 42.394926). The slope \( m \) is \( \frac{42.394926 - 23.545363}{32} \approx 0.5881 \). Thus, the linear model is \( L(z) = 0.5881z + 23.545363 \).

Step 5: Comparing Models

Compare the cubic model \( f(z) \) and the linear model \( L(z) \). While the linear model provides a simpler representation and captures the overall increase in recipients, it may not account for any variations in growth rate over the years, which the cubic model can capture. Thus, the cubic model might be more realistic for capturing detailed variations and trends.

Key Concepts

Medicare Recipients

Medicare is a program designed to provide health insurance to individuals aged 65 and older in the United States, as well as to certain younger individuals with disabilities. Over the years, the number of Medicare recipients has changed due to multiple factors such as the aging baby boomer population and improvements in healthcare. Tracking the change in the number of recipients over the years can help us understand demographic shifts and healthcare needs. In our exercise, the function presented allows us to approximate how the number of Medicare recipients has grown from 1973 to 2005. By analyzing this data, policymakers and researchers can gain insight into the demand on healthcare systems and plan accordingly for future needs.

Linear Approximation

Linear approximation is a useful technique in mathematics which allows us to simplify complex functions. By approximating a more complex function with a linear one, we can work with a simpler model to find trends or make predictions.In the context of our problem, a linear approximation uses two known data points - the number of Medicare recipients in 1973 and in 2005 - to create a straight line. This line provides an average rate of change over the given period. The equation, \( L(z) = 0.5881z + 23.545363 \), describes this straight line, where the slope \( 0.5881 \) represents the average increase in recipients per year. Though linear approximation makes calculations easier, it cannot capture any nuanced changes or fluctuations that might have occurred, unlike more complex functions.

Graphing Functions

Graphing functions is a fundamental skill in calculus and algebra, used to visualize how a function behaves over a certain domain. By plotting functions on a graph, we can easily interpret the rise and fall of values.For the function given in the exercise, we plotted \( f(z) = -0.000015z^3 - 0.005z^2 + 0.75z + 23.5 \) from \( z = 0 \) to \( z = 32 \). This range corresponds to the years 1973 through 2005, representing the growth trend of Medicare recipients. The curvature of the graph provides insight into how the numbers have changed. While plotting, observing the start point at approximately 23.5 million and the endpoint at about 42.4 million offers a visual validation to the known data and trends discussed in the problem.

Cubic Function

Cubic functions are polynomial functions of degree three, characterized by the general form \( f(x) = ax^3 + bx^2 + cx + d \). These functions can model complex behaviors that linear or quadratic functions cannot.The function given in the exercise is a cubic function, allowing us to capture the nuanced growth trend of Medicare recipients over several years. It accounts for variations in growth rates that a simple linear approximation might miss. Cubic functions may have one or two critical points where the rate of change alters, leading to more informative analysis. Understanding the behavior of such a function through its graph can provide valuable insights into the patterns and trends necessary for making informed decisions about healthcare planning.