Problem 18 - Algebra Form and Function

Question

Let \(g(t)\) give the market value (in \(\$ 1000\) s) of a house in year \(t\). Say what the following statements tell you about the house. \(\frac{g(10)-g(4)}{6}=3\)

Step-by-Step Solution

Answer

Answer: The given equation tells us that from year 4 to year 10, the market value of the house increased by $18,000. The house's market value was growing at an average rate of $3,000 per year during this time period.

Step 1: Analyze the given equation

The equation \(\frac{g(10)-g(4)}{6}=3\) represents the average rate of change of the house's market value from year 4 to year 10. We know that the change in market value is given by \(g(10)-g(4)\), and the number of years in that period is 6. Therefore, we have: \(\frac{change\ in\ market\ value}{change\ in\ time} = 3\)

Step 2: Calculate the change in market value

The given equation tells us that the average rate of change in market value is 3. Thus, we need to find the actual change in the market value of the house over the 6-year period. To do this, multiply the rate of change (3) by the change in time (6): Change in market value = 3 * 6 = 18 This tells us that the market value of the house increased by $18,000 over the 6-year period.

Step 3: Interpret the result

Based on the given information, we find that from year 4 to year 10, the market value of the house increased by \(18,000. This means that the house's market value was growing at an average rate of \)3,000 per year during this time period.

Key Concepts

Change in Market Value

The concept of "change in market value" is crucial in understanding how a house's value evolves over time. In this exercise, the change in market value refers to the difference between the house's value at two different points in time, specifically year 10 and year 4. This change is represented mathematically by subtracting the market value at year 4 from the market value at year 10, denoted as \( g(10) - g(4) \). This value informs us about how much the house's market value has increased or decreased over a specific period.

To get a clearer picture, we utilize the formula for the average rate of change, which in this case, given by the exercise, is 3. Multiplying this rate by the time interval of 6 years gives us a total increase of 18 (in terms of \( \(1,000s\) dollars). Thus, the house's market value grew by \( \\)18,000 \) over these 6 years.

Understanding the change in market value is essential for homeowners, real estate agents, and investors to make informed decisions regarding property sales and investments. It provides a numeric basis for evaluating market trends and price movements.

Market Value Function

A market value function, like \( g(t) \) in this problem, expresses the value of an asset, such as a house, at different points in time. It serves as a mathematical model that can represent various economic or financial behaviors over time. In practical terms, knowing what this function looks like can help predict future market values or assess past financial decisions.

In the context of this exercise, \( g(t) \) provides a snapshot of the house's market value measured in thousands of dollars for each year \( t \). By examining specific points on this function, such as \( g(4) \) and \( g(10) \), it becomes possible to calculate changes in value over any given period. This feature is particularly helpful for analyzing long-term investments in real estate and evaluating property performance.

Furthermore, having a functional expression of market values enables more complex analyses, like forecasting future price trends, assessing risk, or evaluating the influence of external economic factors on property prices.

Interpretation of Results

Interpreting results, like those obtained from the average rate of change calculation in this problem, provides deep insights into the long-term value trends of assets. The result of our calculations tells us not just how much the house's market value increased but also the rate at which this growth occurred. In this scenario, the average annual increase of \( \\(3,000 \) signifies steady growth.

Such information can be pivotal for making strategic financial decisions. For a homeowner, understanding that their property is appreciating at \( \\)3,000 \) annually may impact their choices on whether to sell, invest further in the property, or hold on for even greater future appreciation. For investors and market analysts, this growth rate can help compare the property against other investment opportunities.

Furthermore, recognizing patterns from these interpretations helps in assessing whether the growth aligns with broader market trends or if it reflects specific factors related to the property or its location. It's these insights that empower stakeholders to make more grounded, data-driven decisions regarding real estate investments.