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Graph sketching of exponential functions starts with understanding the behavior. For the function \( y=\left(\frac{1}{3}\right)^{x} \), you need to observe how its values change as \( x \) varies. Firstly, recognize the starting point at \( (0,1) \). This is because any number raised to the zero power is 1. Then, gather more key points like \( (1, \frac{1}{3}) \) and \( (-1, 3) \), which serve as a guide for plotting the curve.
Connect these points smoothly to form the graph. The exponential decay will lead the graph to trail closer to the x-axis as \( x \) increases, but never touch it. When sketching, keep in mind the symmetry and smoothness typical of exponential graphs.

An asymptote is a line that a graph approaches but never actually reaches. For the exponential function \( y=\left(\frac{1}{3}\right)^{x} \), the horizontal asymptote is \( y=0 \).
This means as \( x \) increases indefinitely, the value of \( y \) gets closer and closer to 0 but never truly becomes 0. In your graph, represent this with a dashed line along the x-axis. This visual cue helps ensure the plotted function doesn't falsely intersect or touch the x-axis.

• Important: The function's exponents being negative due to its base being less than 1 is what keeps the graph decreasing and hugging the x-axis thus creating an asymptote at \( y=0 \).

Identifying key points is crucial for an accurate sketch of any graph. For \( y=\left(\frac{1}{3}\right)^{x} \), start with the base point \( (0,1) \), then calculate additional points by substituting diverse values of \( x \).
As detailed:

• For \( x=1 \), the point is \( (1, \frac{1}{3}) \).
• For \( x=-1 \), the point becomes \( (-1, 3) \).

Key points demonstrate the graph's direction and steepness. Once plotted, these points allow for the curve to take its natural exponential form, showing its decreasing trend as \( x \) increases.

Exponential functions with bases less than 1, like \( y=\left(\frac{1}{3}\right)^{x} \), are called decreasing functions. This means that as \( x \) increases, the function value \( y \) decreases. Graphically, this appears as a downward slope moving from left to right.
Why does this happen? As the exponent \( x \) grows positively, multiplying the fraction \( \frac{1}{3} \) by itself results in smaller numbers. So, the higher \( x \) gets, the closer \( y \) approaches zero. Yet, the reverse happens when \( x \) is negative, since raising a fraction to a negative power yields larger values. The function's decreasing nature is due to this consistent reduction in value as \( x \) progresses positively.