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Graphing exponential functions is a visual way to understand the behavior of these types of mathematical expressions. An exponential function has the form \( y = a^x \). In this case, \( a \) is the base of the exponent and determines the function's behavior.
For the function \( y = \left( \frac{1}{5} \right)^x \), we are dealing with an exponential decay function because the base \( \frac{1}{5} \) is less than 1.

To graph this function, start by calculating a few key points by plugging in values of \( x \) into the function. This gives you coordinates \( (x, y) \), which you can then plot on a graph.

• When \( x = 0 \), \( y = 1 \).
• When \( x = 1 \), \( y = \frac{1}{5} \).
• When \( x = -1 \), \( y = 5 \).

These points start to shape the curve of the graph.
Once you plot enough points, draw a smooth curve through them, letting the graph naturally approach the x-axis but rarely, if ever, touching it. This method will reveal the exponential curve, highlighting its unique characteristics, such as the horizontal asymptote at \( y = 0 \).

Exponential decay occurs in certain exponential functions where the base \( a \) is a fraction (i.e., between 0 and 1).
This results in a function that decreases rapidly at first and then more slowly over time, approaching zero but never quite reaching it.

In the example function \( y = \left( \frac{1}{5} \right)^x \), each time \( x \) increases by 1, \( y \) is multiplied by \( \frac{1}{5} \).
This sees a rapid decrease in the value of \( y \) as you move from left to right across the graph.

• The decay is steepest around values of \( x \) close to zero.
• The graph approaches zero as \( x \) becomes more positive.
• The graph will rise sharply as \( x \) moves into negative territory.

The decay creates a curve that is useful in modeling real-world phenomena like radioactive decay or depreciation of assets.
Recognizing and interpreting exponential decay is important for understanding how quickly or slowly processes happen over time.

Plotting on a coordinate plane is fundamental for visualizing any function, especially exponential ones, due to their unique curved shapes.
The coordinate plane consists of two perpendicular lines (axes): the x-axis (horizontal) and the y-axis (vertical).

To plot points from an exponential function like \( y = \left( \frac{1}{5} \right)^x \):

• Choose a range of \( x \) values, both positive and negative, to capture the behavior of the function.
• Calculate the corresponding \( y \) for each \( x \) to get a list of coordinate pairs \( (x, y) \).
• Plot each point on the coordinate plane using its coordinates, where the first number represents the position on the x-axis and the second on the y-axis.
• For this function, points might include \( (-2, 25), (-1, 5), (0, 1), (1, \frac{1}{5}), (2, \frac{1}{25}) \).

Once plotted, join these points with a smooth, continuous curve.

This visual can help you see how the function decreases and where it levels off, never completely reaching the x-axis due to the horizontal asymptote.