Problem 4 Graph. $$ y=(0.2)^{x} $$... [FREE SOLUTION] (2024)

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Graphing is the process of representing equations or data points visually on a coordinate plane. For your exponential function, this involves plotting points that result from inputting different x-values into your equation. Once plotted, these points can be connected smoothly to illustrate how the function behaves.

In our case, we are graphing the function \( y = (0.2)^{x} \). Start by finding some key points by calculating the y-values for various x-values. Then use these points to create your graph. As you plot more points, the shape of the graph becomes clearer, helping you understand the function's increasing or decreasing trends.

A coordinate plane is a two-dimensional surface formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). Each point on the plane is defined by a pair of numbers known as coordinates. The first number is the x-coordinate, and it tells how far left or right a point is from the y-axis, while the second number is the y-coordinate, indicating how far up or down it is from the x-axis.

To graph the equation \( y = (0.2)^{x} \) on a coordinate plane, you plot the points that you calculated from your table of values. Each point, such as \( (-2, 25) \), tells you exactly where to place a dot on your graph: 2 units to the left of the y-axis and 25 units above the x-axis.

Creating a table of values is a technique used in graphing to determine key points of a function. You start by selecting a range of x-values and then compute the corresponding y-values using the given equation.

• Select x-values: Choose a few x-values within a practical range, such as \( x = -2, -1, 0, 1, 2 \).
• Calculate y-values: Insert each x-value into the equation \( y = (0.2)^x \).
• Record the values: Create a table to organize these (x, y) pairs.

For example:

• \( x = -2, y = (0.2)^{-2} = 25 \)
• \( x = -1, y = (0.2)^{-1} = 5 \)
• \( x = 0, y = (0.2)^{0} = 1 \)
• \( x = 1, y = (0.2)^{1} = 0.2 \)
• \( x = 2, y = (0.2)^{2} = 0.04 \)

Now, use these points to plot your graph accurately.

A decreasing function is a function where the y-value diminishes as the x-value increases. For the equation \( y = (0.2)^x \), since the base (0.2) is less than 1, the function will decrease.

For example, when \( x = -2 \), the y-value is 25. As \( x \) moves towards positive values (e.g., 1), the y-value decreases to 0.2.

This behavior can be observed by plotting the points and connecting them. The graph will start from a high point on the left and decline sharply, approaching the x-axis as x increases. Hence, it's crucial to understand that the smaller the base of an exponential function, the more rapidly it decreases.