Problem 48 - 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. $$ y=9\left(\frac{2}{5}\right)^{t} $$

Step-by-Step Solution

Answer

This model represents exponential decay, not growth. The decay factor of this model is \(\frac{2}{5}\).

Step 1: Identifying the type of exponential model

Consider the function \(y=9\left(\frac{2}{5}\right)^{t}\). Here, the base of the exponent is \(\frac{2}{5}\), which is less than 1. Thus, this model represents exponential decay.

Step 2: Identifying the decay factor

In an exponential decay function, the decay factor is simply the base of the exponent. So, in this function the decay factor is \(\frac{2}{5}\).

Step 3: Graphing the model

To graph this model, use the general form of the exponential function \(y=ab^{t}\), where 'a' represents the initial value and 'b' is the base. Our 'a' is 9 and 'b' is \(\frac{2}{5}\). Start by plotting the point (0,9) because at t=0, y=a. Then consider different time points (t), calculate their corresponding y-values, and plot them. Over time with a decay factor less than 1, the graph will be a decreasing curve lying in the first and second quadrants with a horizontal asymptote at y=0.

Key Concepts

Exponential Function

An exponential function is a type of mathematical function that grows or decays at a rate proportional to its current value. This can be represented as \( y = ab^{t} \), where:

• \( y \) is the final amount.
• \( a \) is the initial value or starting point.
• \( b \) is the base or growth/decay factor.
• \( t \) is the time variable or exponent.

In exponential decay, as illustrated by the function \( y=9\left(\frac{2}{5}\right)^{t} \), the base \( b \) is less than 1. This feature implies the value of \( y \) decreases over time. It is essential to distinguish between growth and decay. If \( b \) is greater than 1, it signifies exponential growth, meaning the quantity is increasing over time. If \( b \) is between 0 and 1, it describes exponential decay, as seen in this exercise.

Decay Factor

The decay factor in an exponential function is the number that multiplies the initial value, reducing it over time. In the formula \( y=ab^{t} \), \( b \) represents this factor. For a function like \( y=9\left(\frac{2}{5}\right)^{t} \), the decay factor is clearly \( \frac{2}{5} \).
This factor is crucial because it determines how fast the exponential decay happens. The closer \( b \) is to zero, the quicker the decline of the function's value.
Knowing the decay factor helps predict future values by seeing how the current amount decreases over each period. Always ensure to identify the decay factor to understand the rate at which the value drops.

Graph of Exponential Function

When graphing an exponential function, there are key elements to consider for a clear visual representation. The general form \( y=ab^{t} \) helps us plot the function accurately:

• Start by plotting the point \((0, a)\), where \( t=0 \), because \( y=a \).
• Choose several values of \( t \) to calculate corresponding \( y \) values, forming a pattern on the graph.
• In exponential decay, like with \( y=9\left(\frac{2}{5}\right)^{t} \), the curve will slope downward from left to right.
• The graph approaches a horizontal asymptote along the x-axis, but never quite touches it.

This visual representation shows the characteristic decrease of values over time and helps understand how the function behaves in practical scenarios, giving a complete picture of the change occurring.