Problem 44 - Algebra 1: Concepts and Skills

Question

Classify the model as exponential growth or exponential decay. Then identify the growth or decay factor and graph the model. $$ y=24(1.18)^{t} $$

Step-by-Step Solution

Answer

The given model represents exponential growth with a growth factor of 1.18. Therefore, the quantity increases by 18% each time unit.

Step 1: Identify the Model Type

Examine the given model. In this case, we have an equation of the form \(y = ab^t\), where a is the initial value, b is the base, and t is time. If the base b is greater than 1, the model depicts exponential growth. If the base b is between 0 and 1, the model depicts exponential decay. The given model has a base of 1.18, which is greater than 1, hence, the model represents exponential growth.

Step 2: Identify the Growth Factor

The growth factor in an exponential growth model is (b - 1) where b is the base of the model. In this case, the base of the model is 1.18. Hence the growth factor = \(1.18 - 1 = 0.18\). This means the quantity increases by 18% each time unit.

Step 3: Graph the Model

The graph of an exponential growth model is a curve that rises from left to right. At t = 0, the point (0, a) is on the graph, where a is the initial value in the equation \(y = 24(1.18)^t\). Therefore, the point (0, 24) is on the graph. As t increases, the value of y keeps increasing. Thus, the graph starts from (0, 24) and rises to the right.

Key Concepts

Exponential Functions

Exponential functions describe situations where a quantity grows or decays at a rate proportional to its current value. This is mathematically represented as y = ab^{t}, where y is the final amount, a is the initial amount, b is the base or growth (if >1) or decay (if <1) factor, and t is time. For instance, population growth, radioactive decay, and interest compounded continuously are phenomena that can be modeled using exponential functions.

These functions are unique compared to linear functions because they involve the base raised to a variable exponent. This fundamentally changes the way the function behaves, driving either a growth or decay that accelerates over time, unlike the constant rate change seen in linear relationships.

Graphing Exponential Models

Graphing an exponential function is key to understanding its behavior. To graph y = ab^{t}, you start by plotting the initial value (0, a) on the coordinate plane. As the variable t increases, if b > 1, the graph will rise rapidly from left to right, reflecting exponential growth. Conversely, if 0 < b < 1, the graph will decrease from left to right, indicating exponential decay.

When graphing, note crucial points, especially where the graph intercepts the axes. As t approaches infinity, the y-value will also approach infinity if the function represents growth, or approach zero in case of decay. Knowing how to accurately plot these points and draw the curve is essential for visualizing and analyzing the behavior of exponential functions.

Growth Factor Identification

Identifying the growth factor in an exponential function y = ab^{t} is straightforward but crucial for understanding the rate of change. The growth factor is found by extracting the base b and comparing it to 1. If b > 1, the growth factor is b - 1, which means the rate of increase in percentage is (b - 1) * 100%. If b represents decay (meaning 0 < b < 1), the decay factor would be 1 - b, representing the percentage decrease.

In the example y = 24(1.18)^{t}, the base is 1.18. Since it is greater than 1, we're dealing with growth, and the growth factor is 0.18, meaning the quantity is growing by 18% each time period. Understanding the growth or decay factor helps in predicting future values and is fundamental in applications like finance and population studies.