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Graphing exponential functions involves understanding the basic characteristics and plotting points to create a smooth curve. An exponential function like \[ f(x) = \left(\frac{2}{5}\right)^{x} \] exhibits certain predictable behaviors that help us graph it efficiently.**Finding Key Points**New graphers often begin by finding key points like the y-intercept, which always occurs at \( (0, 1) \) for a standard function of the form \( a^x \), as \( a^0 = 1 \). Additionally, choosing a few more x-values helps in sketching:

• Choose an x-value like 1 or 2 to see how quickly the function decreases.
• Choose a negative x-value to observe the growth to the left.

**Plotting and Connecting**after finding these points, plot them on a coordinate graph. Draw a smooth curve through these points, understanding that it approaches the horizontal asymptote (discussed later) as x gets larger.

Exponential decay is a key concept when dealing with functions like \( f(x) = \left(\frac{2}{5}\right)^{x} \) where the base is a fraction less than 1.**Characteristics of Decay**The graph of such a function decreases as x increases:

• The function slowly approaches zero, never actually reaching it.
• As x moves to negative values, the function's values increase because you're essentially taking the reciprocal when you raise it to a negative power.

**Real-World Examples**You might see exponential decay in natural processes such as radioactive decay, where the amount of substance decreases over time in a manner that resembles our function.

Asymptotes are lines that the graph of a function approaches but never quite touches or crosses.**Horizontal Asymptote of Exponential Functions**For the function \( f(x) = \left(\frac{2}{5}\right)^{x} \), the horizontal asymptote is the x-axis, or the line \( y = 0 \). This occurs because as x increases, the function's values decrease towards zero.**Behavior at Infinity**

• As x approaches positive infinity, \( f(x) \) gets very close to zero, but doesn't reach it, ensuring the curve levels off parallel to the x-axis.
• This horizontal line represents a limit the function never exceeds or equals in an exponential decay scenario.

Understanding these asymptotes gives insight into the long-term behavior and boundaries of the graph.