Problem 18 - Algebra Form and Function

Question

Identify the initial value and the rate of change, and explain their meanings in practical terms. The value of an antique is \(2500+80 n\) dollars, where \(n\) is the number of years since the antique is purchased.

Step-by-Step Solution

Answer

Answer: The initial value of the linear equation is $2500, which represents the cost of the antique when it is first purchased. The rate of change is $80 per year, which represents the increase in the value of the antique each year. In practical terms, this means that the antique is initially worth $2500, and its value grows at a steady rate of $80 per year.

Step 1: Identify the Initial Value

To find the initial value, we need to determine the value of the antique when \(n = 0\). Here, the equation is given as: \(Value = 2500 + 80n\). When \(n=0\), the equation becomes \(Value = 2500\). So, the initial value of the antique is $2500.

Step 2: Identify the Rate of Change

The rate of change of this linear equation can be found by examining its slope. The slope is the coefficient of the 'n' term, which in our case is \(80\). This means that for every additional year, the value of the antique increases by $80.

Step 3: Explain the Practical Significance of Initial Value and Rate of Change

Now that we have identified the initial value (\(2500) and the rate of change (\)80), we can discuss their practical meanings. The initial value tells us the cost of the antique when it is first purchased or at the starting point (0 years). So, the antique is initially worth \(2500. The rate of change represents the increase in the value of the antique per year. So, every year after the antique is purchased, its value increases by \)80. In practical terms, this means that as time goes on, the value of the antique grows at a steady rate of \(80 per year, and its initial worth is \)2500.

Key Concepts

Initial Value

The initial value in a linear equation is the starting point before any changes take place. Think of it as the initial amount or the base value before the effects of time or any other factor come into play. In our example of the antique, the equation is given as \(Value = 2500 + 80n\).

To find the initial value of this linear function, we need to consider what happens when \(n = 0\), which represents the number of years since the purchase. When \(n = 0\), the equation simplifies to \(Value = 2500\). This means that when the antique is first purchased, it has an initial value of \(\$2500\).

In practical terms, the initial value signals the worth of the item at the outset. This concept is crucial in understanding any linear equation because it provides a baseline from which changes can be measured. It tells us the starting price before any increase due to the passage of time or other factors is added.

Rate of Change

The rate of change is a critical concept in linear equations, reflecting how one quantity changes in relation to another. Essentially, it tells us the speed or pace of change over time or other increments. In mathematical terms, the rate of change is often represented by the slope of the linear equation. In the antique value equation \(Value = 2500 + 80n\), the rate of change is \(80\).

This number tells us that for each additional year \(n\), the value of the antique increases by \(\$80\). The concept is especially useful because it provides a constant measure of how much more valuable the antique becomes every year.

Understanding the rate of change allows us to predict future values and understand the dynamic nature of the situation being modeled. It's a constant increment that explains how one variable affects another in the equation, helping us see and calculate the progression of value increase.

Slope

The slope in a linear equation is a unit that signifies the rate of change between two variables. In straightforward terms, it represents the steepness or incline of the line on a graph depicting the relationship between these variables. In our antique value equation \(Value = 2500 + 80n\), the slope is the coefficient of \(n\), which is \(80\).

The slope tells us that each unit increase in \(n\) results in an increase of \(\$80\) in the value of the antique. Graphically, it shows how the line rises as \(n\) increases. A steeper slope would indicate a faster increase in value. The slope is a crucial part of understanding linear relationships as it quantifies the rate of change.

By grasping the concept of slope, students can visualize how quickly or slowly one variable changes in relation to another, providing deep insights into the dynamics presented in the equation.