Problem 95 - College Algebra with Modeling and Visualization

Question

U.S. Home Ownership The general trend in the percentage \(P\) of homes lived in by owners rather than renters between 1990 and 2006 is modeled by $$P(x)=0.00075 x^{2}+0.17 x+44$$ where \(x=0\) comesponds to \(1990, x=1\) to \(1991,\) and so on. Determine a function \(g\) that computes \(P\), where \(x\) is the actual year. For example, \(P(0)=44,\) so \(g(1990)=44\)

Step-by-Step Solution

Answer

The function \(g(\text{year}) = 0.00075(\text{year} - 1990)^2 + 0.17(\text{year} - 1990) + 44\).

Step 1: Define the Relationship Between x and Year

In the given problem, the year 1990 corresponds to \(x = 0\), 1991 to \(x = 1\), and so on. Thus, we have the relationship \(x = \text{year} - 1990\).

Step 2: Express x in Terms of the Actual Year

Using the relationship from Step 1, rewrite \(x\) as a function of the year: \(x = \text{year} - 1990\). This allows us to convert any given year into the corresponding \(x\) value used in the function \(P(x)\).

Step 3: Substitute x in Terms of Year into P(x)

Substitute the expression for \(x\) from Step 2 into the function \(P(x)\): \[ P(x) = 0.00075x^2 + 0.17x + 44 \] becomes \[ P(\text{year} - 1990) = 0.00075(\text{year} - 1990)^2 + 0.17(\text{year} - 1990) + 44 \]

Step 4: Define Function g for Given Year

The function \(g\) that computes \(P\) for a given year can now be defined as:\[ g(\text{year}) = 0.00075(\text{year} - 1990)^2 + 0.17(\text{year} - 1990) + 44 \] This expression calculates the percentage \(P\) when given the actual year.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers of variables. They are of the form: \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \), where \(a_n, a_{n-1}, \ldots, a_1, a_0\) are constants. The highest power of the variable \(x\) is called the degree of the polynomial.
For example, in the expression \(P(x) = 0.00075x^2 + 0.17x + 44\), we have a polynomial of degree 2. This is because the highest power of \(x\) is 2.
Such polynomials are often referred to as quadratic functions. Quadratic functions can create parabolic graphs, which are U-shaped curves that can open upwards or downwards based on the leading coefficient. In our example, the graph will open upward because the coefficient 0.00075 is positive.
Polynomials can model various real-world situations by providing an approximate representation of data trends over time or other variable changes. Understanding these functions allows us to predict and analyze trends effectively.

Function Transformation

Function transformation involves shifting, stretching, compressing, or flipping graphs of functions. It is an essential aspect of understanding how changes in equations affect their graphs.
The transformation process allows us to adapt functions to suit different contexts or convert them from one form to another. This is particularly useful in making models more relevant to real-world scenarios.
To understand function transformation, consider \(P(x) = 0.00075x^2 + 0.17x + 44\). The transformation mainly occurs through shifting, which includes moving graphs horizontally or vertically.

• Horizontal Shifts: Occur when \(x\) is replaced by \(x - h\). Here, shifting \(x\) to \(\text{year} - 1990\) indicates a horizontal shift, aligning our function with the actual years 1990 onwards.
• Vertical Shifts: Involve adding or subtracting constants from the function, affecting how high or low the graph is placed on the coordinate plane.
• Stretching or Compressing: These affect the graph's shape, where the factors involve multiplying \(x\)-values or \(y\)-values by a constant.
• Reflection: This transformation flips a graph over an axis, altering its orientation.

Translating a function like converting \(x\) to "year - 1990" allows us to understand data concerning actual time values, making it very practical for everyday use.

Mathematical Modeling

Mathematical modeling uses mathematical languages and constructs to represent real-world phenomena. It plays a critical role in predicting outcomes, analyzing trends, and making decisions in fields like economics, engineering, and social sciences.
The function \(P(x) = 0.00075x^2 + 0.17x + 44\) serves as a model for homeownership trends. By setting \(x\) based on the year, we can calculate the percentage of homes lived in by owners for any year between 1990 and 2006.
Creating a model involves:

• Identifying variables: Determine what factors affect the phenomenon and how they interact. Here, \(x\) represents the years since 1990.
• Establishing relationships: Develop equations that relate the variables. Our example establishes a quadratic relationship.
• Using data: Collect data to fit and refine the model. The equation's coefficients are determined from historical data.
• Analysis and prediction: Use the model to understand trends or predicts future results, such as housing market trends.

Models are simplifications of reality. They help us gain insights into complex systems by using mathematical theory and statistical methods, making them an indispensable tool in problem-solving and decision-making.