Problem 83 - 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 \(f\). (Refer to the introduction to this section.) The table lists total iPod sales \(y\) in millions \(x\) years after 2004 $$ \begin{array}{cccc} x & 0 & 2 & 4 \\ y & 3 & 55 & 150 \end{array} $$

Step-by-Step Solution

Answer

The solution is \( a = 5.375 \), \( b = 15.25 \), \( c = 3 \). Predicted sales for \( x = 6 \) is calculated using \( f(6) \).

Step 1: Set Up the System of Equations

We are given the quadratic function model: \( f(x) = ax^2 + bx + c \). We have three data points: \((0, 3), (2, 55), (4, 150)\). For each point, substitute \(x\) and \(y\) values into the equation to form a system of equations.1. For \( (0, 3) \): Substitute \( x = 0\) and \( y = 3 \), we get: \[ 3 = a(0)^2 + b(0) + c \] \[ c = 3 \]2. For \( (2, 55) \): Substitute \( x = 2\) and \( y = 55 \), we get: \[ 55 = a(2)^2 + b(2) + c \] \[ 55 = 4a + 2b + c \]3. For \( (4, 150) \): Substitute \( x = 4\) and \( y = 150 \), we get: \[ 150 = a(4)^2 + b(4) + c \] \[ 150 = 16a + 4b + c \].Now, our system of equations is:1. \( c = 3 \)2. \( 4a + 2b + 3 = 55 \)3. \( 16a + 4b + 3 = 150 \).

Step 2: Solve the System of Equations

Using the first equation \( c = 3 \), substitute \( c \) into the other equations:1. \( 4a + 2b + 3 = 55 \) becomes \( 4a + 2b = 52 \).2. \( 16a + 4b + 3 = 150 \) becomes \( 16a + 4b = 147 \).Now, the new system is:1. \( 4a + 2b = 52 \)2. \( 16a + 4b = 147 \).Divide the second equation by 2 for convenience: \( 8a + 2b = 73.5 \).Subtract the first equation from this: \( (8a + 2b) - (4a + 2b) = 73.5 - 52 \)\( 4a = 21.5 \)\( a = 5.375 \).Plug \( a = 5.375 \) into \( 4a + 2b = 52 \): \( 4(5.375) + 2b = 52 \)\( 21.5 + 2b = 52 \)\( 2b = 30.5 \)\( b = 15.25 \).The solution is \( a = 5.375 \), \( b = 15.25 \), \( c = 3 \).

Step 3: Use Technology for Verification

Use a calculator or software like Python, MATLAB, or a graphing calculator to solve the system:1. Confirm that \( a = 5.375 \), \( b = 15.25 \), and \( c = 3 \) satisfy the equations.2. Verify descending order of variable coefficients through substitution back into the original system of equations for more accurate validation.

Step 4: Graph the Function and Data Points

Plot the quadratic function \( f(x) = 5.375x^2 + 15.25x + 3 \).1. Use graphing software or a calculator to visualize this curve.2. Plot the given data points: \((0, 3)\), \((2, 55)\), and \((4, 150)\) on the same set of axes.3. Ensure the curve passes roughly through or near these points to validate the model.

Step 5: Make a Prediction

Use the derived function to make a prediction:1. Choose a value for \( x \) that represents a year after 2004, for example, \( x = 6 \) (corresponding to the year 2010).2. Calculate \( f(6) = 5.375(6)^2 + 15.25(6) + 3 \).3. Compute and interpret the result, which represents the predicted iPod sales in millions for that year.

Key Concepts

System of Equations

A quadratic function is often represented as \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants to be determined. To find these values, we use what is called a "system of equations." Essentially, we collect equations that share variables and solve them simultaneously to find the values of these variables.

When given data points, like our iPod sales data

• ((0, 3), (2, 55), (4, 150)), we assign each pair of values (x, y) into the quadratic equation to form three separate equations.

Each of these equations when substituted gives a condition that must be true for the solution. The first point

• ((0, 3)) leads to \( c = 3 \) because when \( x = 0 \), the \( ax^2 \) and \( bx \) terms disappear.
• The point (2, 55) gives us the equation \( 4a + 2b + c = 55 \), contributing another equation to our system.
• Finally, the point (4, 150) results in \( 16a + 4b + c = 150 \).

The next step is solving these equations for the variables \( a \), \( b \), and \( c \), often employing substitution or elimination methods. Substituting \( c = 3 \) into the other equations simplifies them by reducing the number of unknowns, allowing us to solve more easily.

Graphing Functions

Graphing functions is a crucial part of understanding how a formula models real-life data. After we determine the values of \( a \), \( b \), and \( c \) for our quadratic function, the next step is to visualize it.

This visualization helps us see how well the function fits the data points.

• We plot the calculated quadratic function \( f(x) = 5.375x^2 + 15.25x + 3 \).
• Using graphing software or a graphing calculator, draw the curve that this function describes.This could involve software like Desmos or graphing calculators, which render these plots accurately.
• Next, add the original data points \((0, 3)\), \((2, 55)\), and \((4, 150)\) on the same axes.

Observing the graph, we confirm whether the curve passes close to or directly through these points. The closer the curve aligns with these points, the more accurately our function describes the data. This direct comparison highlights any discrepancies and helps verify our calculations.

Predictive Modeling

Predictive modeling involves using an established trend to make forecasts about future data points. In our exercise, we constructed a predictive model using the quadratic function obtained from the iPod sales data. This model can predict sales for years beyond those given.

To make predictions:

• Choose a future value of \( x \) representing years after 2004. For instance, \( x = 6 \), corresponding to the year 2010.
• Substitute this value into our equation \( f(x) = 5.375x^2 + 15.25x + 3 \).
• Calculate \( f(6) = 5.375(6)^2 + 15.25(6) + 3 \) to find the predicted number of iPods sold.

The result provides an estimate of iPod sales, purely based on the pattern observed in former years. This method of predictive modeling illustrates how mathematical equations not only explain historical data but also empower us to anticipate future trends.