Problem 96 - College Algebra with Modeling and Visualization

Question

The function \(D\) defined by $$D(x)=2375 x^{2}+5134 x+5020$$ models AIDS deaths \(x\) years after 1984 . Write a formula \(g(x)\) that computes AIDS deaths during year \(x,\) where \(x\) is the actual year.

Step-by-Step Solution

Answer

The formula is \( g(x) = 2375(x - 1984)^2 + 5134(x - 1984) + 5020 \).

Step 1: Identify the Relationship

We are given that the function \(D(x)\) models AIDS deaths \(x\) years after 1984. This means if \(x=0\), it corresponds to the year 1984. For any year \(x\), we need to find the equivalent \(x\) years after 1984.

Step 2: Define the New Function Variable

To convert a given year into a variable usable by \(D(x)\), subtract 1984 from the actual year. If \(x\) is the year, then the equivalent variable for \(D\) is \(x - 1984\).

Step 3: Create Function g(x)

Substitute \(x - 1984\) into \(D(x)\). Thus the formula for \(g(x)\), representing AIDS deaths in the actual year \(x\), is: \[ g(x) = D(x - 1984) = 2375(x - 1984)^2 + 5134(x - 1984) + 5020 \]

Step 4: Simplify g(x)

To simplify \(g(x)\), expand \((x - 1984)^2\): \( (x - 1984)^2 = x^2 - 2 \cdot 1984 \cdot x + 1984^2 \). Then substitute this into the equation: \[ g(x) = 2375(x^2 - 2 \cdot 1984 \cdot x + 1984^2) + 5134(x - 1984) + 5020 \]Expand and simplify further if needed.

Key Concepts

Quadratic Functions

Quadratic functions are a fundamental concept in algebra that describe a wide range of natural phenomena. A quadratic function is typically written in the form:

• \( f(x) = ax^2 + bx + c \)

where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\) to ensure the function is quadratic. These functions create parabolas when graphed, which can open upwards or downwards depending on the sign of \(a\).

In our exercise, the function \(D(x) = 2375x^2 + 5134x + 5020\) is a quadratic function that models AIDS deaths over the years since 1984. In this function:

• \(a = 2375\)
• \(b = 5134\)
• \(c = 5020\)

This function tells us that the number of deaths follows a parabolic trend, influenced by the passage of years since 1984. Understanding how to work with quadratic functions is essential for interpreting such real-world data, as they are often used to model various types of growth and decay.

Polynomial Expansion

Polynomial expansion involves expressing a polynomial term raised to a power as a sum of terms. This process is often used to simplify complex algebraic expressions and make them more understandable. In the context of our problem, we need to expand the expression

• \((x - 1984)^2\)

This expansion will help us write a simplified version of the function \(g(x)\) that models the actual year.

The binomial expansion of \((x - 1984)^2\) is accomplished as follows:

• \((x - 1984)^2 = x^2 - 2 \times 1984 \times x + 1984^2\)

Calculating each term, this becomes:

• \(x^2 - 3968x + 3936256\)

Substituting this expanded form back into the function \(g(x)\) allows for further simplification, converting a more general model to one applicable in specific contexts. Polynomial expansions simplify evaluations and are crucial in algebraic manipulation.

Modeling with Functions

Modeling with functions is a powerful technique in mathematics used to represent real-world situations. The exercise involves using the quadratic function \(D(x)\) to model AIDS deaths starting from 1984. This model transforms the problem of understanding trends over the years by leveraging the mathematical properties of the function.

• To convert years to the model, the exercise requires calculating \(x - 1984\), which adjusts the year to the zero point of 1984.
• This makes it possible to use the given data to predict the number of deaths in a particular year, providing valuable insights into the disease's impact over time.

These transformations make functions highly adaptable tools—whether used for simple trend analysis or complex predictive modeling. The ability to convert complex information into understandable patterns allows for better planning and decision-making. Understanding these transformations is key to effective modeling and sheds light on past and future trends expressed numerically through functions.