Medicare recipients The function f given by f(x)=-0.000015...

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?

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?

Key Concepts

Model Comparison

Model comparison is a process of evaluating different mathematical models to determine which one best fits the data and most realistically describes the underlying phenomenon. This involves assessing aspects such as complexity, interpretability, and the accuracy in capturing behavior across the range of the data. It often requires considering both statistical measures of fit and the practical implications of the model in the real world.

Real-World Data Modeling

Real-world data modeling involves creating mathematical representations of actual phenomena, taking into account the variability and potential nonlinearity inherent in the data. This process requires balancing complexity with simplicity to develop models that are both accurate and interpretable. In practice, the chosen model should not only fit the historical data well but also provide realistic predictions under reasonable assumptions.

Parameter Transformation

Parameter transformation involves shifting or scaling the variable in a function to simplify calculations or interpretation. In many modeling contexts, a common practice is to redefine the independent variable (for instance, subtracting a constant) to center the data. This can improve numerical stability and make the coefficients in the model more interpretable relative to a specific baseline or starting point.

Polynomial Function Modeling

Polynomial functions, especially cubic functions, are often used to capture nonlinear trends in data over a period of time. They allow for curvature in the model that can accommodate changes in the rate of increase or decrease in the data. This is useful when the underlying relationship in the data is not strictly linear, and higher order terms help in capturing accelerations or decelerations of the trend.

Linear Model Approximation

Linear models assume a constant rate of change and are frequently used for simplicity and ease of interpretation. They provide a straight-line approximation of data trends, which is especially useful when the relationship between the variables is approximately linear or when a simplified summary of the trend is desired. Despite their simplicity, linear models can be less accurate when the actual data behavior is nonlinear.

Graphical Analysis

Graphical analysis is a key method in mathematics for visually assessing the behavior of functions. By plotting functions, one can observe trends, identify key features such as intercepts and turning points, and compare different models. It assists in understanding how well a mathematical model fits the data and in communicating the trends in a clear and intuitive way.

Transcript

00:02 Okay, so in this particular problem, we're given a function that represents the number of medicare recipients.

00:11 And the function, the original function, is given as, i'm not going to read that out, but it's given, the first thing i want to do is graph it.

00:19 So it's given as this top graph here, negative 1, negative 15, what is that, 100 ,000 c cube minus 5 ,000 c squared, and so forth and so on.

00:31 And so hopefully you know by looking at this function with your study of polynomial functions that this is a cubic function, so you know the general shape of the graph is going to be like that.

00:42 The coefficient, the leading coefficient is negative, so it's going to start high and end up going low.

00:49 You know the y intercept is that constant, 23 .5.

00:58 And then you could calculate if you needed to or had to, you could see your roots are there.

01:05 But all we're doing is we're looking at since z is the number of years, or x is the number of years since 1973, we're only looking at a certain portion of that.

01:17 So we're going to zoom in here.

01:18 And really what we're only looking at is this part of the graph right here.

01:22 So you can see when you overlay, it's the same graph, but we're only.

01:26 Focusing in on this section of the graph here.

01:29 So you know the y intercept is 23 .5 and the problem does say that there are about 23 .5 recipients in 1973 and then it goes up to 42 ,394 in the year 2005 about 42 .266 million.

01:52 And notice this is in millions.

01:58 That's why these are decimal numbers.

02:02 So in considering what the number of medicare recipients is doing over time, it's very clear that it's increasing over time.

02:12 If you look at this function, you can see that it's increasing, and you would expect it here to begin to decrease, but we're only looking at it here, so we're not going to talk about any of the decrease because the function that we're limited to here is increasing.

02:28 So the number of medicare recipients is changing over time by increasing.

02:36 So it's increasing.

02:39 And then we can this almost, even though the graph is a cubic function, it almost looks when we zoom in on the part that we're considering, that graph almost looks linear.

02:52 So we're going to take this graph.

02:55 And come up with a linear model for this particular function.

03:01 And in order to do that, the first thing we want to do is find the slope.

03:04 We can pick just two basic points...

Please give Ace some feedback