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Linear growth occurs when a quantity increases by the same fixed amount each period. This type of growth can be represented using a straight line equation, which makes it quite straightforward to analyze and predict outcomes. In our exercise, the home price increased from \(\$100,000\) in 1985 to \(\$200,000\) in 2005, a span of 20 years. To find the yearly increase, we apply the formula: \[ \text{Yearly Increase} = \frac{200,000 - 100,000}{2005 - 1985} = 5,000 \] This means the home's price increased by \(\$5,000\) each year. Thus, the linear growth model is: \[ f(t) = 100,000 + 5,000t \] where \(t\) represents the number of years after 1985. So at \(t = 10\) years (1995), the home price would be \(f(10) = 100,000 + 5,000 \times 10 = 150,000\). This simple relationship helps to easily predict future values.

Exponential growth occurs when the rate of increase is proportional to the current value, leading to growth that accelerates over time. It can be described with the equation: \[ g(t) = P \times (1 + r)^t \] where \(P\) is the initial amount, \(r\) is the growth rate, and \(t\) is time. For the home price, given \( g(20) = 200,000 \) and \( P = 100,000 \), we need to find \( r \). By solving: \[ 200,000 = 100,000 \times (1 + r)^{20} \] We get: \[ 2 = (1 + r)^{20} \] Taking the 20th root on both sides gives: \[ 1 + r = 2^{ \frac{1}{20} } \approx 1.03526 \] Thus, \( r \approx 0.03526 \) or 3.526\%. The exponential growth model becomes: \[ g(t) = 100,000 \times 1.03526^t \] This model implies the price grows faster over time, as the rate compounds.

Modeling functions are mathematical expressions that represent real-world situations. They allow us to predict future behavior based on current data.
In this exercise, we used both linear and exponential models to showcase different types of growth over time.
The linear model \( f(t) = 100,000 + 5,000t \) shows a constant rate of increase, making it easy to predict future values. It's useful for scenarios where changes are consistent and predictable.
The exponential model \( g(t) = 100,000 \times 1.03526^t \) captures compounding growth, which is more realistic for many real-world phenomena, like population growth, inflation, and, in our case, housing prices. Exponential growth reflects how percentages can create larger increments over time.
These models highlight the importance of understanding different types of growth and choosing the appropriate one to make informed predictions.