Graph each function. ƒ(x) = (1/3)^x

Verified step by step guidance

1. Recognize that the function given is an exponential function of the form \(f(x) = a^x\), where the base \(a = \frac{1}{3}\) is a positive number less than 1. This means the graph will represent exponential decay.
2. Create a table of values by choosing several values for \(x\), such as \(-2\), \(-1\), \(0\), \(1\), and \(2\). For each \(x\), calculate \(f(x) = \left(\frac{1}{3}\right)^x\) using the property of exponents.
3. Recall that for negative exponents, \(a^{-x} = \frac{1}{a^x}\). So, for example, \(f(-1) = \left(\frac{1}{3}\right)^{-1} = 3\), and \(f(-2) = \left(\frac{1}{3}\right)^{-2} = 9\).
4. Plot the points from your table on the coordinate plane. For example, plot \((0,1)\) since any number to the zero power is 1, and plot the other points accordingly.
5. Draw a smooth curve through the points, showing the exponential decay behavior: the graph approaches the \(x\)-axis (horizontal asymptote at \(y=0\)) but never touches it, and increases rapidly as \(x\) becomes more negative.

Key Concepts

Exponential Functions

An exponential function has the form f(x) = a^x, where the base a is a positive constant. The function grows or decays depending on whether a is greater than or less than 1. Understanding this helps in predicting the shape and behavior of the graph.

Graphing Exponential Functions

Graphing involves plotting points by substituting values of x into the function and connecting them smoothly. Key features include the y-intercept at (0,1) and the horizontal asymptote, usually the x-axis, which the graph approaches but never touches.

Exponential Decay

When the base of an exponential function is between 0 and 1, the function represents exponential decay. This means the function decreases as x increases, approaching zero but never becoming negative, which affects the direction and shape of the graph.