Graph each function. $\text{ƒ}\left(x\right)=(1/6{)}^{-x}$

Verified step by step guidance

1. Recognize that the function is given by \(f(x) = \left(\frac{1}{6}\right)^{-x}\). The negative exponent means you can rewrite the function using the property of exponents: \(a^{-x} = \frac{1}{a^x}\).
2. Rewrite the function as \(f(x) = \left(\frac{1}{6}\right)^{-x} = 6^x\). This simplifies the function to an exponential function with base 6.
3. Identify key points to plot by choosing values for \(x\), such as \(x = -2, -1, 0, 1, 2\), and calculate the corresponding \(f(x)\) values using \(f(x) = 6^x\).
4. Plot the points on the coordinate plane and observe the shape of the graph. Since the base 6 is greater than 1, the graph will show exponential growth, increasing rapidly as \(x\) increases.
5. Draw the curve through the plotted points, noting that the graph passes through \((0,1)\) because any nonzero number raised to the zero power is 1, and the graph approaches the x-axis but never touches it (asymptote at \(y=0\)).

Key Concepts

Exponential Functions

An exponential function has the form f(x) = a^x, where the base a is a positive real number not equal to 1. These functions model growth or decay processes and have distinctive graphs that either increase or decrease rapidly depending on the base and exponent.

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent, i.e., a^{-x} = 1 / a^x. Understanding this helps rewrite and simplify expressions, which is essential for graphing functions like f(x) = (1/6)^{-x}.

Graphing Exponential Functions

Graphing involves plotting points by substituting values of x and understanding the behavior of the function, such as asymptotes and intercepts. For f(x) = (1/6)^{-x}, recognizing it as (6)^x helps predict an increasing exponential graph with a horizontal asymptote at y=0.