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Exponential decay occurs when a quantity decreases at a consistent rate proportional to its current value. This can be observed in the function \( y = \left(\frac{1}{3}\right)^x \). Here, the base of the exponential is \( \frac{1}{3} \), a value less than 1, indicating the function will decay or decrease as \( x \) increases.
In an exponential decay graph, the function starts with larger values when \( x \) is negative, and as \( x \) becomes positive, the values of \( y \) rapidly decrease. The function approaches, but never quite reaches, the \( x \)-axis, illustrating what's known as an asymptotic behavior.
Exponential decay models real-world scenarios like radioactive decay and cooling of substances. These graphs generally move rapidly towards zero, demonstrating a continuous and rapid fall-off in value.

Plotting key points is a crucial step in graphing any function, particularly for exponential functions where the behavior of the curve can be surprising.
Here’s how you can plot key points for \( y = \left(\frac{1}{3}\right)^x \):

• Start by choosing several \( x \) values that encompass both negative and positive numbers. This gives a broader view of how the function behaves.
• For each chosen \( x \) value, calculate \( y \) using the exponential function. For example:
  • When \( x = -2 \), \( y = 9 \)
  • When \( x = -1 \), \( y = 3 \)
  • When \( x = 0 \), \( y = 1 \)
  • When \( x = 1 \), \( y = \frac{1}{3} \)
  • When \( x = 2 \), \( y = \frac{1}{9} \)

These calculated points provide a framework for sketching your graph, and more points can be used for greater accuracy.

Graph sketching for an exponential function requires connecting plotted key points with a smooth line. Here's how to ensure accuracy in your sketch:

• First, mark the calculated points precisely on the coordinate plane. This spatial relationship is crucial to creating the correct curve.
• Right after that, look at the trend those points form. For \( y = \left(\frac{1}{3}\right)^x \), you’ll observe a steep decrease as you move from left to right.
• Draw a smooth curve through the points starting from the positive regions on the \( y \)-axis. Ensure the curve approaches the \( x \)-axis asymptotically, reflecting the nature of exponential decay.
• Avoid sharp angles or straight lines between points. Exponential functions exhibit smooth, continuous curvature.

After sketching, verify your work using a graphing calculator to ensure the hand-drawn graph accurately reflects the function's expected trend and shape.