Problem 92 - College Algebra with Modeling and Visualization

Question

Use transformations of graphs to model the table of data with the formula \(f(x)=a(x-h)^{2}+k .\) (Answers may vary.) Number of titles released for DVD rentals $$ \begin{array}{llllll} \text { Year } & 1998 & 1999 & 2000 & 2001 & 2002 \\ \text { Titles } & 2049 & 4787 & 8723 & 14,321 & 21,260 \end{array} $$

Step-by-Step Solution

Answer

The quadratic function is \( f(x) = -1668.5(x-2000)^2 + 8723 \).

Step 1: Analyze the Data

We are given data for the number of DVD titles released over a set of years. The years are 1998 to 2002, and the number of titles corresponds to each year. Our goal is to fit a quadratic function to this dataset using transformations of the graph.

Step 2: Determine the Structure of the Quadratic Function

The quadratic function is of the form \(f(x) = a(x-h)^2 + k\). We need to find appropriate values for \(a\), \(h\), and \(k\) that fit the data given. We start by considering the year as \(x\) and the titles as \(f(x)\). We will need to perform a transformation to align our data with this quadratic model.

Step 3: Perform a Horizontal Transformation

We take the middle year for simplicity as \( x = 2000\), which simplifies our calculations. We set \(h = 2000\) in the equation \(f(x)=a(x-h)^2+k\) to represent this as the vertex of the parabola. This means we are centering our quadratic function around the year 2000.

Step 4: Compute Parameters Using a Point

Using the data point for 2000, where \(f(2000) = 8723\), we set \(x = 2000\) in the equation. This indicates that \(k = 8723\), because at \(x = h\), \(f(x) = k\).The equation now simplifies to \(f(x)=a(x-2000)^2+8723\).

Step 5: Calculate the Slope Parameter "a"

Choose an additional data point, such as 1998 with 2049 titles to estimate \(a\). Substitute \(x = 1998\) and \(f(x) = 2049\) into the equation:\[ 2049 = a(1998-2000)^2 + 8723 \]Solving for \(a\), we have:\[ 2049 = 4a + 8723 \]\[ 4a = 2049 - 8723 \]\[ a = \frac{-6674}{4} = -1668.5 \]

Step 6: Formulate the Quadratic Function

Now we have the values for all parameters. The quadratic function that models the given data is:\[ f(x) = -1668.5(x-2000)^2 + 8723 \]

Step 7: Verify and Adjust (if necessary)

Check this function with other data points like 1999, 2001, and 2002. If the function accurately predicts the number of titles (within an acceptable error range), it is a good model. If not, adjustments to \(a\) might be necessary based on more data points or using methods like polynomial regression for better fit.

Key Concepts

Transformations of Graphs

Graph transformations are techniques used in algebra and geometry to manipulate the shape or position of a graph on a coordinate plane. By changing certain parameters of the function, we can predict and model various behaviors in real-world situations. In the context of quadratic functions, these transformations help us to model datasets, like the one concerning DVD title releases, using an equation of the form \(f(x)=a(x-h)^2+k\).

Here's a brief overview of each transformation:

• Horizontal Shifts: The vertex form of a quadratic, \((x-h)^2\), represents a parabola shifted \(h\) units horizontally. Setting \(h = 2000\) centers the parabola on the year 2000, making it the vertex.
• Vertical Shifts: Adding \(k\) to the function moves the whole graph up by \(k\) units. In this exercise, \(k\) represents the number of DVD titles in the year 2000.
• Reflection and Dilation: The coefficient \(a\) affects whether the parabola opens upwards or downwards (reflection) and how wide or narrow it is (dilation). A negative \(a\) found here alters our model to reflect the downturn from the year 2000.

To fit the DVD data, such transformations are used to map the model to actual data points, capturing the trend of DVD releases over the given years.

Data Modeling

Data modeling in mathematics refers to the process of using equations to represent and analyze real-world data. It's crucial when you have datasets and wish to identify patterns or predict future outcomes. In this exercise, we are modeling the number of DVD titles released over a span of years with a quadratic equation.

To model data effectively, there are key steps:

• Identify the type of function that best fits the dataset. Here, a quadratic function is used due to the dataset's nature and assumed consistency with other similar growth scenarios.
• Define your variables. In our case, the year is \(x\), and the number of titles is \(f(x)\). This choice aligns with typical modeling structures.
• Adjust the model according to specific data points. Centering the model around \(x = 2000\) by choosing \(h = 2000\) simplifies computation and visual representation.

By constructing a quadratic model, we can analyze trends—for example, how releases change over time, identify peaks, or plan for future projections if similar behaviors are expected in other media types.

Polynomial Regression

Polynomial regression is a statistical method to model the relationship between a dependent variable and one or more independent variables using polynomial equations. It is especially useful when data shows curvature (non-linear patterns) that a straight line (linear regression) can't capture. In this exercise, choosing a quadratic polynomial allows us to capture the upward trend and slight downturn in the DVD title data.

Steps for polynomial regression include:

• Choose the Degree: Deciding on the polynomial degree is crucial. A quadratic function was appropriate here since we need a parabolic shape to fit the data.
• Fit the Model: Use least squares or other methods to estimate coefficients \(a\), \(h\), and \(k\). This involves solving equations for the best fit, which minimizes error across all data points.
• Validate the Model: Once a model is created, check its accuracy with other data points not used in making the model. Readjust coefficients if necessary to enhance the model's predictive power.

By using polynomial regression, we can create models that offer insights into data that show non-linear relationships. Here, it helped model the rapid growth trend of DVD releases and depicted its eventual decline with precision.