Question

Profit The annual profits \(y\) (in millions of dollars) of UnitedHealth Group from 1997 to 2006 can be modeled by \(y=58.86 x^{2}+228.0 x+677, \quad-3 \leq x \leq 6\) where \(x\) represents the year, with \(x=6\) corresponding to 2006\. (Source: UnitedHealth Group) (a) Find the \(y\) -intercept of the graph of the model. What does it represent in the given situation? (b) Use the model to complete the table of values for \(x=-3,-2,-1,0,1,2,3,4,5\), and 6 (c) Plot the solution points given by the table in part (b) and use the points to sketch the graph of the model.

Short answer

The y-intercept of the graph of the model is 677, which represents the annual profit in millions of dollars in the year 2000. The completed table of values is obtained by inserting each x value into the given quadratic function. The graph of the model is a parabola based on the plotted points.

Step-by-step solution

1. Finding the y-intercept

The y-intercept of a function is the y-coordinate where the graph intersects the y-axis. This can be found by setting \(x = 0\) in the function and evaluating. Substituting \(x = 0\) in \(y = 58.86x^2 + 228.0x + 677\), we get \(y = 677\). The y-intercept of the graph of the model is 677. This represents the annual profit in millions of dollars of UnitedHealth Group for the year when \(x = 0\) i.e., the profit in 2000.

2. Filling the table with model values

The requested values must be computed by plugging in the specified x-values into the model function. To illustrate, \(x= -3\) is inserted into \(y = 58.86x^2 + 228.0x + 677\), yielding \(y = 58.86*(-3)^2 + 228.0*(-3) + 677 = -149.14 \). This process is repeated for the other x values.

3. Plotting the solution points and sketching the graph

The solution points derived from the calculation in step 2 are plotted on a set of axes with x-values ranging from -3 to 6. After plotting these points, they are connected to form the graph of the model. The graph will be visualized as a parabola due to the quadratic term in the function. The location and shape of the graph will be determined by the coefficients of the function.

Key concepts

Y-Intercept of Quadratic Function

Understanding the y-intercept of a quadratic function is fundamental in business algebra, and it provides valuable insight into the initial value of a given situation. In the context of the UnitedHealth Group's profit model, the y-intercept is discovered by setting the independent variable, which is time represented by \(x\), to zero. When \(x=0\), the equation simplifies to \(y=58.86x^2 + 228.0x + 677\), which yields \(y=677\).

This y-intercept value of 677 represents the annual profit in millions of dollars for the base year in the model, which could correlate to the year 2000 if \(x=0\) coincides with that year. This starting point is crucial for businesses, as it reflects the initial financial state before any changes over time that the quadratic function may describe.

In our exercise, the y-intercept serves as a cornerstone for graphing the entire function and helps to understand the company's financial standing at the outset of the time series analyzed.

Table of Values for Quadratic Function

A table of values is an effective way to organize and compute the results of a quadratic function for specific inputs. To compile a comprehensive table for the UnitedHealth Group's annual profits, we plug different values of \(x\), ranging from -3 to 6, into the profit model \(y=58.86x^2 + 228.0x + 677\). Each calculation corresponds to a different year and demonstrates the expected annual profit.

For instance, when \(x=-3\), we insert this value into the quadratic equation to find \(y=58.86(-3)^2 + 228.0(-3) + 677\), which simplifies to a specific profit figure for that year. Repeating this process for each \(x\)-value in the given range allows us to observe the changing profits over time and lays the groundwork for visual analysis through graphing. The resulting table not only provides specific data points but also helps identify trends and patterns in the profit model.

Graphing Quadratic Functions

Graphing quadratic functions is a pivotal skill in business algebra, as it visually represents complex relationships. The graph of a quadratic function like the profit model of UnitedHealth Group is a parabola. To sketch this curve, we plot the points obtained from the table of values on a coordinate grid.

Starting with the y-intercept at 677, we then plot additional points for each year, positioning them accurately based on their calculated profits. Once all the points from \(x=-3\) to \(x=6\) are placed on the grid, we draw a smooth curve connecting them to form the characteristic 'U' shape of the parabola.

The graph's direction, whether it opens upward or downward, is determined by the coefficient of the \(x^2\) term. In this case, a positive coefficient causes the parabola to open upwards, indicating that profits increase until reaching a maximum point and then start to diminish. This visual representation provides a powerful tool for businesses to predict trends, identify maximum and minimum values, and make strategic decisions based on the graph's trajectory.