Problem 31 - Algebra 1: Concepts and Skills

Question

Graph the exponential decay model. $$ y=15(0.9)^{t} $$

Step-by-Step Solution

Answer

The graph of the exponential decay model \(y = 15(0.9)^{t}\) begins at a higher value and reduces over time. The calculated points are:(-1, 16.67), (0, 15), (1, 13.5), (2, 12.15), (3, 10.84), (4, 9.844)

Step 1: Identify Parameters of the Exponential Decay Model

Observe the given exponential decay model \(y = 15(0.9)^{t}\). Here \(a = 15\) represents the initial value, and \(b = 0.9\) (less than 1) represents the decay factor per time period \(t\).

Step 2: Calculate Coordinates for the Graph

Choose values for \(t\) and calculate corresponding values for \(y\) using the given formula. To illustrate, let's choose \(t = -1, 0, 1, 2, 3, 4\). Calculate the corresponding values of \(y\): For \(t = -1\), \(y = 15(0.9)^{-1} = 16.67\), For \(t = 0\), \(y = 15(0.9)^{0} = 15\), For \(t = 1\), \(y = 15(0.9)^{1} = 13.5\), For \(t = 2\), \(y = 15(0.9)^{2} = 12.15\), For \(t = 3\), \(y = 15(0.9)^{3} = 10.94\), For \(t = 4\), \(y = 15(0.9)^{4} = 9.844\)

Step 3: Plotting the Points

These pairs \((-1, 16.67), (0, 15), (1, 13.5), (2, 12.15), (3, 10.84), (4, 9.844)\) now represent points that can be plotted on the graph to portray the exponential decay.

Step 4: Draw the Exponential Decay Curve

Having plotted the points calculated in previous step, you can now draw the curve passing through the points to show the trend of decay. The curve should start from a higher point and gradually come down, symbolizing decay.

Step 5: Analyzing the Graph

By analysing the graph, one can see that as time \(t\) increases, the value of \(y\) decreases, which is what we should expect from an exponential decay model. This shows that the initial quantity diminishes over time in a non-linear manner, characterized by a constant ratio of change.

Key Concepts

Initial Value

When we talk about the initial value in an exponential decay model, it's like setting the starting line for a race. In the given equation \(y = 15(0.9)^{t}\), the initial value is represented by the constant \(a = 15\). This initial value indicates the amount or level you start with before any decay begins. It's crucial because it defines the baseline from which all subsequent changes are measured. Imagine pouring a new drink into a cup: the drink's initial volume is similar to this initial value.
In practical terms, the initial value answers the questions like "How much did we start with?" or "What was the initial population size?". It assists in projecting and predicting how quickly or slowly the decay process will affect the initial amount over time.

Decay Factor

The decay factor is what dictates the pace of the reduction. It's a bit like controlling the speed at which you turn the volume down on your music player. In the context of our equation \(y = 15(0.9)^{t}\), the decay factor is the number \(0.9\).

• If the decay factor is less than 1, like in this example, it means you're dealing with a situation where the quantity decreases over time.
• The decay factor tells us how much the exponential function reduces the initial value for each time unit \(t\) we increase.
• The smaller the decay factor, the faster the initial value decreases.

It's essential in scenarios like radioactive decay, population decline, or value depreciation of assets like cars and electronics. Understanding this factor helps predict when and by how much the quantity in question will shrink.

Graphing

Graphing an exponential decay model transforms numbers into a visual story. When you graph the equation \(y = 15(0.9)^{t}\), you're plotting how the initial value of 15 changes over different time intervals \(t\). You start by calculating several points that lie on the curve by plugging in values for \(t\). For instance, at \(t = 0\), the value of \(y\) is 15, and as \(t\) increases, \(y\) diminishes based on the decay factor.
Here are some steps to create the curve:

• Choose several values of \(t\), like -1, 0, 1, 2, 3, etc.
• Calculate the corresponding \(y\) values based on your formula.
• Plot these \(t, y\) coordinate pairs on a graph.

This plotted curve typically starts high and slowly descends. The graph helps visualize the decay's nature, showing it as a smooth decline rather than a rapid drop. Observing the curve can help you understand not just how quickly, but how consistently, the initial value is diminishing over time. It offers a clear insight into the entire decline process, making abstract numbers tangible.