Question: Draw the graph of \( f(x) = \left(\frac{1}{6}\right)^x \). [Graph grid with x-axis from -10…

Draw the graph of \( f(x) = \left(\frac{1}{6}\right)^x \).

[Graph grid with x-axis from -10 to 10 and y-axis from -10 to 10]

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Solution

To draw the graph of the function \( f(x) = \left(\frac{1}{6}\right)^x \), follow these steps: Identify the base of the exponential function. The base is \(\frac{1}{6}\), which is a fraction less than 1, indicating exponential decay. Find a few key points to plot. Calculate the function value for a few values of \( x \): For \( x = 0 \): \[ f(0) = \left(\frac{1}{6}\right)^0 = 1 \] For \( x = 1 \): \[ f(1) = \left(\frac{1}{6}\right)^1 = \frac{1}{6} \] For \( x = 2 \): \[ f(2) = \left(\frac{1}{6}\right)^2 = \frac{1}{36} \] For \( x = -1 \): \[ f(-1) = \left(\frac{1}{6}\right)^{-1} = 6 \] For \( x = -2 \): \[ f(-2) = \left(\frac{1}{6}\right)^{-2} = 36 \] Plot these points on the graph. The points \( (0, 1) \), \( (1, \frac{1}{6}) \), \( (2, \frac{1}{36}) \), \( (-1, 6) \), and \( (-2, 36) \) will help you sketch the curve. Draw the curve. Connect the points smoothly, forming a curve that approaches the x-axis as \( x \) increases and rises steeply as \( x \) decreases (since the base is less than 1, the function decays). Ensure the curve passes through all points and maintains the behavior of an exponential decay function.