Problem 19 - Precalculus with Limits

Question

Graphing an Exponential Function In Exercises \(17-22,\) use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. $$f(x)=6^{-x}$$

Step-by-Step Solution

Answer

The graph of the function \(f(x) = 6^{-x}\) is an exponential decay curve that approaches the x-axis as \(x\) goes to positive infinity, and skyrockets to positive infinity as \(x\) goes to negative infinity.

Step 1: Create a Table of Values

Create a table to show some values of \(x\) and the resulting values of \(f(x) = 6^{-x}\). Choose a reasonable range of \(x\) values such as -5, -2, 0, 1, 2, 3. Calculate \(f(x)\) by substituting each chosen \(x\) value into \(f(x) = 6^{-x}\).

Step 2: Plot the Points

After computing the function values, list the points \((x, f(x))\) on a graph. Each pair corresponds to a point in the form \((x, y)\), where \(x\) is the horizontal coordinate and \(f(x) = 6^{-x}\) is the vertical coordinate. Plot each point on a coordinate system.

Step 3: Draw the Graph

Once all points \((x, f(x))\) are plotted, draw a smooth curve through the points to represent the function \(f(x) = 6^{-x}\). Pay attention to how the graph behaves as \(x\) approaches positive and negative infinity, because this will help in sketching the overall curve of the function.

Key Concepts

Understanding Graphing

Graphing an exponential function like \( f(x) = 6^{-x} \) involves visualizing how the function behaves across various values of \( x \). This process helps to convey important insights about the function's relationship between the independent variable \( x \) and the dependent variable \( f(x) \).

When graphing, it is essential to understand the nature of the function. An exponential function, where the variable \( x \) appears as an exponent, typically exhibits rapid change. For \( f(x) = 6^{-x} \), as \( x \) increases, \( f(x) \) decreases and approaches zero. Conversely, as \( x \) decreases, the value of \( f(x) \) increases. This results in a characteristic curve that approaches the x-axis but never quite touches it.

• Choose a range for \( x \) values to ensure you capture the behavior of the function across both negative and positive values.
• Visualize how the curve starts from a high point on the left (for negative \( x \)), then falls towards the x-axis as \( x \) becomes positive.

Graphing not only provides a visual picture but also aids in interpreting other properties like intercepts and asymptotes.

The Coordinate System Explained

A coordinate system is the backdrop on which graphs are drawn. In most cases, particularly with functions of one variable like \( f(x) = 6^{-x} \), we use a two-dimensional coordinate system consisting of an x-axis and a y-axis perpendicular to each other.

The x-axis is the horizontal axis and represents the input values, or the domain of the function. The y-axis is the vertical axis and represents the output values, or the range of the function. Each point on the plane is defined by a pair of coordinates \((x, y)\). Here, \( x \) is the input value and \( y \) is the corresponding function value \( f(x) \).

Using the coordinate system, you can:

• Plot points by determining their location along the x-axis and y-axis.
• Understand the interaction between variables through their placement.
• Draw graphs by connecting plotted points, visualizing the relationship between \( x \) and \( f(x) \).

The coordinate system is crucial for understanding how the function behaves across different intervals and helps in determining features like growth, decay, or constancy displayed by the graph.

Creating a Table of Values

Making a table of values is a fundamental step in graphing functions, particularly exponential ones like \( f(x) = 6^{-x} \). This table helps you organize your work by listing certain \( x \) values and the corresponding function values \( f(x) \).

To create it effectively, choose a reasonable range of \( x \) values based on the behavior you want to capture. For \( f(x) = 6^{-x} \), considering values like -5, -2, 0, 1, 2, and 3 can be useful. For each \( x \) value, substitute into the function to calculate \( f(x) \). The computations for this function would look something like this:

• If \( x = 0 \), then \( f(x) = 6^{0} = 1 \).
• If \( x = 1 \), then \( f(x) = 6^{-1} = \frac{1}{6} \).
• As \( x \) increases, \( f(x) \) becomes smaller, showing the inverse relationship.

This table acts as your guide when plotting points on the graph. It helps ensure that you accurately represent how \( f(x) \) changes with \( x \). This step is critical for sketching an accurate graph.