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Exponential decay describes a process where a quantity decreases at a constant percentage rate over time. In mathematical terms, an exponential decay function takes the form \( y = a \, b^{-x} \), where:

• \( y \) is the output or dependent variable.
• \( a \) is the initial amount or starting value.
• \( b \) is the base of the exponential function, which should be greater than 1 for it to represent decay.
• \( x \) is the independent variable, often representing time.

In an exponential decay, as \( x \) (for example, time) increases, the value of \( y \) decreases because the factor \( b^{-x} \) makes the function value smaller. In the given function \( y = 0.2 \, (10^{-x}) \), \( a = 0.2 \) and \( b = 10 \), indicating that the function decays rapidly as \( x \) increases.

To visualize an exponential decay function on a graph, you need to understand a few basic steps about graphing functions. Graphing involves plotting a series of points from an equation on a coordinate plane and then connecting them to form a curve or line.For instance, suppose you want to graph \( y = 0.2 \, (10^{-x}) \). First, calculate some points using different \( x \) values. Then, plot these points accurately on the graph. Connecting the points smoothly gives a visual representation of how the function behaves. For exponential decay, expect the graph to drop sharply as \( x \) increases.

A table of values is a helpful tool used to organize results for different inputs of an equation. This aids in plotting the graph of a function. To create a table of values for an exponential decay function like \( y = 0.2 \, (10^{-x}) \), follow these steps:

• Choose a range of \( x \) values. Common choices might include negative integers, zero, and positive integers.
• Substitute each \( x \) value into the function to find the corresponding \( y \) values.

For the function given, choosing values such as \( x = -2, -1, 0, 1, 2 \) can simplify calculations and ensure clarity. This table allows you to see how quickly \( y \) decreases as \( x \) rises, confirming the exponential decay behavior.

A coordinate plane is a two-dimensional surface where you can visually represent mathematical equations. It consists of two perpendicular number lines: the horizontal axis (x-axis) and the vertical axis (y-axis). The intersection of these axes is called the origin, indicated by the point (0, 0).When graphing the function \( y = 0.2 \, (10^{-x}) \) on a coordinate plane, each point \((x, y)\) represents a specific solution to the equation. Plot these points using the values obtained from your table. Each point corresponds to a pair of coordinates that show the relationship between \( x \) and \( y \). As you connect the plotting points, the curve should demonstrate a rapid decline from left to right, showcasing the nature of exponential decay on the graph. This visual understanding can help in comprehending how changes in \( x \) affect \( y \).