91Ó°ÊÓ

In this exercise, we graph the exponential function \(f(x) = 6^{-x}\). Exponential functions are unique due to their rapid change in value, either growing or decaying, depending on the exponent.
Here’s how to graph an exponential function step-by-step:

• Identify the base and exponent.
• Create a table of values by selecting x-values and calculating the corresponding y-values.
• Plot the (x, y) points on a graph.
• Connect the points with a smooth curve.

This specific function has a base of 6 and a negative exponent, resulting in a decay function. Observe how the y-values change dramatically even with small adjustments to x, reflecting the rapid decay nature.
To highlight this, consider the points we calculated: (-2, 36), (-1, 6), (0, 1), (1, 1/6), and (2, 1/36). When we plot these, they form a curve that descends fast and approaches the x-axis but never quite touches it. This is characteristic of the exponential decay.

Exponential decay describes a process where the quantity decreases rapidly at first, then levels off over time. The general form of an exponential decay function is \(f(x) = a \times b^{-x}\), where \(0 < b < 1\) or, equivalently, \(f(x) = a \times \frac{1}{b^x}\) if \(b > 1\).
In our exercise, \(f(x) = 6^{-x}\), the base 6 is greater than 1 but the negative exponent turns it into a decay function.
Exponential decay is significant in various real-life contexts, such as:

• Radioactive decay - where substances decline over time.
• Cooling of objects - where temperature drops exponentially until it stabilizes at ambient temperature.
• Depreciation of assets - where the value of an asset decreases rapidly over a period before becoming negligible.

When graphing exponential decay, expect to see a curve that starts high on the left and declines rapidly, approaching zero but never touching the x-axis.

Understanding the properties of negative exponents is crucial for working with exponential functions. A negative exponent signifies the reciprocal of the base raised to the corresponding positive exponent.
For instance, \(6^{-x}\) means \( \frac{1}{6^x} \). This represents how as the input value increases, the output value decreases exponentially.
Here are some key properties and rules to remember:

• \(a^{-n} = \frac{1}{a^n}\) - The negative exponent rule.
• Application in various functions like in reciprocals: \(f(x) = 2^{-x} = \frac{1}{2^x}\)

Negative exponents are particularly useful for solving equations involving dividing terms and converting them to a more manageable form. They frequently appear in scenarios like calculating decay, fractions, and scientific notation.
In exercises like ours, knowing that \(6^{-x}\) equates to \(\frac{1}{6^x}\) simplifies the process of calculating y-values and understanding the graph's behavior.