Problem 68 - Algebra A Combined Function

Question

The number of a certain chain of stores \(y\) for the years \(2003-2007\) can be modeled by the equation \(y=-198 x+3991,\) where \(x\) represents the number of years after 2003. (Source: Limited Brands) a. Find the \(y\) -intercept of this equation. b. What does this \(y\) -intercept mean?

Step-by-Step Solution

Answer

The y-intercept is 3991; it represents the number of stores in 2003.

Step 1: Understanding the Equation

The given equation is a linear equation in the form \( y = mx + c \), where \( m \) is the slope and \( c \) is the \( y \)-intercept. The equation provided is \( y=-198x+3991 \).

Step 2: Identify the y-intercept

In any linear equation of the form \( y = mx + c \), the \( y \)-intercept is the constant term \( c \). In the equation \( y=-198x+3991 \), the \( y \)-intercept is \( 3991 \).

Step 3: Interpretation of the y-intercept

The \( y \)-intercept occurs when \( x = 0 \). In the context of the problem, \( x \) represents the number of years after 2003. Therefore, \( x = 0 \) corresponds to the year 2003. The \( y \)-intercept of 3991 indicates the number of stores in the year 2003.

Key Concepts

Slope-Intercept Form

Linear equations can often be expressed in the slope-intercept form. This form is written as \( y = mx + c \), where \( m \) is the slope of the line, and \( c \) is the y-intercept. This format makes it very easy to understand and predict the behavior of a straight line on a graph.
When you have a linear equation, identifying these two components allows you to quickly draw the line or to describe how one variable changes in relation to the other. The slope \( m \) defines the steepness of the line, showing how much \( y \) changes for a unit change in \( x \). The y-intercept \( c \) tells us where the line crosses the y-axis.
In the given exercise, the equation \( y = -198x + 3991 \) is already in slope-intercept form. This reveals that the line has a slope of \(-198\) and a y-intercept of \(3991\). Understanding this form can help you analyze and model real-world situations more effectively.

Y-Intercept

The y-intercept is an essential concept in understanding linear equations. It is the point where the line crosses the y-axis on a graph, which occurs when \( x = 0 \). In mathematical terms, it is represented as the constant \( c \) in the equation \( y = mx + c \).
Finding the y-intercept is straightforward. Simply evaluate the equation when \( x = 0 \). For the equation \( y = -198x + 3991 \), setting \( x = 0 \) gives \( y = 3991 \). This means the line intersects the y-axis at \( 3991 \).
The significance of the y-intercept is that it provides a starting point for the line on a graph. It tells us the initial value of \( y \) when there are no changes in \( x \). This initial value is crucial, especially in real-world applications, as it denotes an initial condition or status at the beginning of your observation.

Mathematical Modeling

Mathematical modeling involves using mathematical structures and equations to represent real-world phenomena. It helps in understanding, analyzing, and predicting behaviors within various systems.
The linear equation \( y = -198x + 3991 \) is a model that relates time to the number of stores in a chain. \( x \) denotes the number of years after 2003, while \( y \) represents the corresponding number of stores.
This model helps project the store numbers for this chain from 2003 onwards. The slope \(-198\) indicates a downward trend, suggesting that the store count decreases by 198 each year. Through modeling, businesses and analysts can make informed predictions and decisions based on historical data fitting into this equation.

Interpreting Functions

Understanding and interpreting functions is a key skill in mathematics. Functions describe how two variables are related. By interpreting the function, we can gather valuable insights into the behavior of one variable as the other changes.
In our function \( y = -198x + 3991 \), it shows the relationship between \( x \) (years after 2003) and \( y \) (number of stores). The downward slope indicates a steady decline in the number of stores over time.
Interpreting such functions allows us to answer questions like, "How many stores were there at the initial year?" or "What trend does the function suggest for future years?" In this example, the y-intercept, \( 3991 \), indicates that the starting number of stores in 2003 was 3991, and the negative slope tells us the chain of stores is decreasing each subsequent year.