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Graphing exponential functions involves drawing a curve for equations where the variable is in the exponent. An exponential function can be written in the form of \( f(x) = a^x \), where \( a \) is a constant base and \( x \) is the exponent. Depending on the value of \( a \), the graph can show either exponential growth (if \( a > 1 \)) or exponential decay (if \( 0 < a < 1 \)).

In the given exercise, the function \( g(x) = \left( \frac{1}{5} \right)^x \) is an example of an exponential decay function because the base \( \frac{1}{5} \) is less than one. To graph this function, one must plot several points based on \( x \)-values and connect them smoothly to show the curve's behavior. The graph will typically have a horizontal asymptote, often along the x-axis, as \( x \) tends toward positive infinity.

Exponential decay refers to the process where quantities diminish at a rate proportional to their current value. This is typified by functions where the base \( a \) is between 0 and 1. They decrease as the x-values increase, forming a downward-sloping curve.

In the function \( g(x) = \left( \frac{1}{5} \right)^x \), each increase in \( x \) results in the function value getting smaller. This reflects the general behavior of exponential decay as the base being less than 1 makes the exponent value decrease from left to right on the graph. This can occur naturally in various situations such as radioactive decay or depreciation of an asset.

Plotting important points is a fundamental step in drawing the graph of a function. By selecting key x-values, calculating corresponding y-values, and marking these on a graph, you create a path the function follows.

For \( g(x) = \left( \frac{1}{5} \right)^x \), choosing integers such as \( x = -2, -1, 0, 1, \) and \( 2 \) provides a clear picture of how the function behaves. Substituting these values into the function, we get:

• For \( x = -2 \), \( g(-2) = 25 \)
• For \( x = -1 \), \( g(-1) = 5 \)
• For \( x = 0 \), \( g(0) = 1 \)
• For \( x = 1 \), \( g(1) = 0.2 \)
• For \( x = 2 \), \( g(2) = 0.04 \)

Plotting these points, and drawing a smooth curve through them, provides the visual representation of how the function decreases.

Transformations of functions include shifting, stretching, compressing, or reflecting a graph. These do not fundamentally alter the type of function but change its position or shape on the graph.

Knowing how transformations affect exponential functions enhances interpretation. For instance, a vertical shift might occur if a constant is added outside the exponential term. Horizontal shifts happen with changes inside the function's exponent. Reflecting occurs with a negative sign in front of the function or exponent.

In \( g(x) = \left( \frac{1}{5} \right)^x \), transformations may be applied to analyze changes in the graph appearance without impacting its nature as an exponential decay function. This involves visual adjustments rather than a formula revision, making it an essential skill for understanding diverse graphical presentations of functions.