Problem 32 - Algebra 1: Concepts and Skills

Question

Graph the exponential decay model. $$ y=72(0.85)^{t} $$

Step-by-Step Solution

Answer

The graph of the exponential decay model \(y=72(0.85)^t\) starts at (0, 72) and decreases as t increases, approaching the x-axis but never reaching it.

Step 1: Identify Initial Amount and Decay Factor

In decay function \(y=72(0.85)^t\), the value 72 is the initial amount (the amount at time zero), and 0.85 is the decay factor per time period.

Step 2: Table of Key Points

Create a table of key points. Plot several values of t (time) ranging from 0 to 5 or so, and calculate the corresponding value of y (amount) using the given decay model. For instance, when \(t = 0\), \(y = 72(0.85)^0 = 72\). This gives the point (0,72) on the graph.

Step 3: Plot Graph

Use these points to sketch a smooth, decreasing graph. This should start at (0, 72) and approach the x-axis as t increases.

Step 4: Label Graph

Label the axes as 'time' (x-axis) and 'amount' (y-axis), and give your graph a title.

Key Concepts

Initial Amount

In an exponential decay model, the initial amount is the starting value before any decay begins. You can think of it as the amount present at time zero. In the equation \(y = 72(0.85)^t\), the term '72' represents this initial amount. It's the baseline from which you measure how much the quantity decreases over time.
Understanding the initial amount is crucial since it sets the context for the overall behavior of the decay process. You can imagine it as your starting point in a race; you can't know how far you've traveled without knowing where you started.

Decay Factor

A decay factor is what determines how much a quantity decreases per time period in an exponential decay model. In our example equation \(y = 72(0.85)^t\), the decay factor is \(0.85\). This means that with each passing unit of time, the quantity becomes 85% of what it was in the previous time period.
Whenever the decay factor is between 0 and 1, it indicates exponential decay. The closer the factor is to zero, the faster the decay process. By understanding the decay factor, you can predict how quickly the initial amount will diminish.
Remember, any value above 1 would indicate exponential growth, not decay.

Graphing Exponential Functions

Graphing exponential decay functions can be an exciting way to visualize the decrease of a quantity over time. To graph a function like \(y = 72(0.85)^t\), you start by plotting the initial amount at \(t = 0\). That's the point (0, 72) on the graph.
Then, by calculating several values of \(y\) for different \(t\) values, you can plot these points to create a graph. You might choose \(t\) values such as 1, 2, 3, etc., calculating each corresponding \(y\) using the decay formula. For instance:

• If \(t = 1\), \(y = 72(0.85)^1 = 61.2\).
• If \(t = 2\), \(y = 72(0.85)^2 = 52.02\).

Connect these points smoothly to form a downward-sloping curve.
This graph demonstrates the concept of decay visually, helping one see how the quantity approximates zero as time progresses. Label your axis for clarity. The x-axis represents time, while the y-axis represents the amount.

Table of Key Points

Creating a table of key points is a great way to systematically organize your calculations when dealing with exponential decay functions. Each row typically corresponds to a different time value \(t\), and you'll calculate the corresponding \(y\) using the given exponential equation.
For the function \(y = 72(0.85)^t\), you might start by considering time values from \(t = 0\) to \(t = 5\). For each \(t\), compute \(y\):

• \(t = 0\), \(y = 72\)
• \(t = 1\), \(y = 61.2\)
• \(t = 2\), \(y = 52.02\)
• \(t = 3\), \(y = 44.22\)
• \(t = 4\), \(y = 37.59\)

This table serves as a data resource to assist in plotting points on your graph, allowing you to understand precisely how the quantity drops over different periods. It’s a helpful tool for ensuring your graph's accuracy.