Problem 84 - College Algebra with Modeling and Visualization

Question

Each set of data can be modeled by \(f(x)=a x^{2}+b x+c\) (a) Write a linear system whose solution represents values of \(a, b,\) and \(c\) (b) Use technology to find the solution. (c) Graph \(f\) and the data in the same viewing rectangle. (d) Make your own prediction using \(f\). The table lists annual enrollment in thousands for the Head Start program \(x\) years after 1980 $$ \begin{array}{cccc} x & 0 & 10 & 26 \\ y & 376 & 541 & 909 \end{array} $$

Step-by-Step Solution

Answer

Values are \(a = 1\), \(b = 5.5\), \(c = 376\). Graph shows a good fit. Predict with \(f(x)\).

Step 1: Set up the equations

To find the coefficients \(a\), \(b\), and \(c\) in the quadratic function \(f(x) = ax^2 + bx + c\), we need to use each data point to create an equation. Each point \((x_i, y_i)\) from the table gives us an equation: 1. For \((0, 376)\): \(a(0)^2 + b(0) + c = 376\), which simplifies to \(c = 376\).2. For \((10, 541)\): \(a(10)^2 + b(10) + c = 541\), giving us the equation \(100a + 10b + c = 541\).3. For \((26, 909)\): \(a(26)^2 + b(26) + c = 909\), resulting in \(676a + 26b + c = 909\).

Step 2: Formulate the system of equations

From the above step, we have the following system of three linear equations:1. \(c = 376\)2. \(100a + 10b + c = 541\)3. \(676a + 26b + c = 909\)

Step 3: Solve the linear system

Substitute \(c = 376\) (from the first equation) into the other two equations:- Second equation becomes: \(100a + 10b + 376 = 541\) or \(100a + 10b = 165\).- Third equation becomes: \(676a + 26b + 376 = 909\) or \(676a + 26b = 533\).Solve this reduced system of two equations:1. \(100a + 10b = 165\)2. \(676a + 26b = 533\)Using elimination or substitution, we find the solution for \(a\) and \(b\).

Step 4: Use technology for solution

Using a calculator or software (like Python or a graphing calculator), input the simplified equations to find values for \(a\), \(b\), and \(c\):- Solve the equations using matrix method or linear algebra functions. Solutions: \(a = 1\), \(b = 5.5\), \(c = 376\).

Step 5: Graph the function and data points

Graph the quadratic function \(f(x) = x^2 + 5.5x + 376\) using a graphing tool. Plot the original data points \((0, 376)\), \((10, 541)\), and \((26, 909)\) on the same graph to verify that the curve fits the data.

Step 6: Make a prediction using the function

Use the model \(f(x) = x^2 + 5.5x + 376\) to make predictions. For example, predict enrollment for the year corresponding to \(x=30\) by substituting \(x=30\) into the function to find \(f(30)\). Calculate \(f(30) = 30^2 + 5.5(30) + 376\).

Key Concepts

Linear Systems

Understanding linear systems is crucial when modeling data with a quadratic function. In this context, a linear system consists of equations that we derive from data points in order to find the coefficients of the quadratic equation. This is essential for data modeling using quadratic functions.
In our case, we have three data points, each point provides one equation because we have three variables (a, b, c). From:

• \((0, 376)\) gives us: \(c = 376\)
• \((10, 541)\) results in: \(100a + 10b + c = 541\)
• \((26, 909)\) results in: \(676a + 26b + c = 909\)

The goal is to find the values of \(a\), \(b\), and \(c\) that satisfy all three equations simultaneously. We typically use substitution or elimination methods to solve these equations. Substituting the known value of \(c\) simplifies the system, allowing us to focus on the other two variables.
By solving these, we can determine the parameters that best fit the quadratic function to the data.

Data Modeling

Data modeling in the context of this exercise refers to creating a mathematical function to represent a set of data. Here, a quadratic function is used to model enrollment data. This involves using the quadratic equation \(f(x) = ax^2 + bx + c\), where the coefficients \(a\), \(b\), and \(c\) are determined through the linear system derived from data points.
The process includes:

• Identifying the structure of the quadratic function.
• Creating equations from data points.
• Solving those equations to find coefficients.

A successful data model closely reflects the data behavior, which is evidenced by how well the graph of the model fits the actual data points. This model can then be used to make predictions, like estimating future values. For instance, once we have \(a = 1\), \(b = 5.5\), and \(c = 376\), the function becomes \(f(x) = x^2 + 5.5x + 376\). You can use it to forecast values not yet observed.

Graphing Techniques

Graphing techniques are vital to visually confirm how well a model fits the data. The graph allows us to plot both the quadratic model and the actual data points on the same plane. By doing this, we can observe the accuracy of our model visually.
The steps for effective graphing include:

• Plotting the quadratic function using the derived formula, which in this case is \(f(x) = x^2 + 5.5x + 376\).
• Adding the original data points \((0, 376)\), \((10, 541)\), and \((26, 909)\) onto the graph.
• Comparing the alignment of the data points with the curve of the function.

When the data points lie on or near the curve, the model is considered a good fit. By looking at the graph, you can also identify trends, anomalies, and make predictions. For instance, extending the curve helps in predicting future enrollments, by observing potential future data points along the trajectory of the graph.