Problem 16 - Algebra Form and Function

Question

Identify the initial value and the rate of change, and explain their meanings in practical terms. The number of people enrolled in Mathematics 101 is \(200-5 y,\) where \(y\) is the number of years since 2004 .

Step-by-Step Solution

Answer

Answer: The initial value, 200, represents the number of people enrolled in Mathematics 101 in 2004. The rate of change, -5, indicates that the number of enrolled students decreases by 5 people per year.

Step 1: Identify the given equation and variables

The given equation is \(200 - 5y\), where \(y\) represents the number of years since 2004.

Step 2: Identify the initial value

The initial value is the number of people enrolled in Mathematics 101 when \(y = 0\) (in 2004). To find this, substitute \(y = 0\) into the equation: \(200 - 5(0) = 200\). The initial enrolment in 2004 is \(200\) people.

Step 3: Identify the rate of change

The rate of change is the coefficient of the \(y\) term in the equation, which is \(-5\). This means that for each passing year, the number of enrolled students decreases by 5.

Step 4: Explain the meanings of initial value and rate of change in practical terms

The initial value, \(200\), represents the number of people enrolled in Mathematics 101 in 2004. The rate of change, \(-5\), indicates that for each year that passes, the number of enrollees decreases by 5 people. If this trend continues, we can expect to see a decline in Mathematics 101 enrolments over time.

Key Concepts

Initial Value

In the context of algebraic expressions, the initial value represents the starting point of a function or equation when the independent variable is zero. In simpler terms, it's the number at the beginning before any changes occur.

For this exercise, the initial value is found by evaluating the expression \(200 - 5y\) at \(y = 0\). This results in \(200\), indicating that 200 people were enrolled in Mathematics 101 in the year 2004. This initial figure is crucial as it sets the baseline from which changes are measured.

This value is often seen in linear equations representing real-world situations, such as population, sales, or any starting data point. By identifying the initial value, you can better understand the situation at the beginning and predict or analyze changes over time.

Rate of Change

The rate of change is a fundamental concept in algebra, particularly in linear equations, representing how one quantity changes in relation to another. It is most commonly expressed as the coefficient of the variable.

In our given equation \(200 - 5y\), the rate of change is \-5\. This tells us that the number of students enrolled in Mathematics 101 decreases by 5 every year. The rate of change allows us to understand the dynamics of the model. It sheds light on how quickly or slowly the numbers are changing over time.

Recognizing the rate of change is vital in predicting future trends and making informed decisions. Whether it is a positive or negative rate, it helps visualize how a situation evolves or devolves over the specified period.

Linear Equation

The equation \(200 - 5y\) is a classic example of a linear equation, which is any equation that forms a straight line when graphed. These equations have variables raised only to the first power, ensuring a constant rate of change.

Linear equations are typically presented in the form \(y = mx + c\), where \(m\) is the slope (rate of change) and \(c\) is the initial value (y-intercept). In our scenario, the expression shows how the number of enrollees changes over time. Here, \(m = -5\) and \(c = 200\).

Using linear equations, we can make forecasts, such as how many students will remain enrolled after a specific number of years. They simplify the analysis of relationships between two variables, providing clear, straightforward ways to predict trends from data.