Problem 17 - 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. $$f(x)=\left(\frac{1}{2}\right)^{x}$$

Step-by-Step Solution

Answer

The graph of the function \(f(x)=(1/2)^x\) is a decreasing curve. Going from the top left of the graph to the bottom right, it never touches the x-axis because \(f(x)\) is always greater than 0 for any real number x.

Step 1: Construct a Table of Values

Using a graphing calculator or online plotting tool, plot \(f(x) = (1/2)^x\) for values of x ranging from -3 to 3. Make a note of the f(x) values that are yielded by these x values.

Step 2: Plot the Points

Using the table of values you have generated in Step 1, plot the points on a graph. For example, the points might be (-3,8), (-2,4), (-1,2), (0,1), (1,0.5), (2,0.25), and (3,0.125).

Step 3: Sketch the Graph

Subsequently, using these points as guides, sketch the complete graph of the function. The graph should start from the top left, descending towards the right, showing that as x increases, f(x) decreases.

Key Concepts

Graphing Utilities

Graphing utilities are fantastic tools to visualize complex mathematical functions like exponential ones. These utilities can be software tools or apps on your calculator that assist in plotting these functions to better understand their behavior. For the exponential function \( f(x) = \left( \frac{1}{2} \right)^{x} \), a graphing utility helps you easily generate a list of points by performing tedious calculations instantly.

Using a graphing calculator or an online tool, you simply input the function, and choose the x-values you wish to analyze. The utility calculates and displays the corresponding y-values, allowing you to identify key characteristics of the function.
Additionally, graphing utilities can help predict future behaviors of the graph even outside your initial scope, making them invaluable for students learning exponential growth and decay functions.

Table of Values

Creating a table of values is a crucial step in understanding any function's behavior, especially exponential functions. By choosing a range of x-values, such as from -3 to 3, and calculating the corresponding y-values, you can see how the function behaves over that interval.

For the function \( f(x) = \left( \frac{1}{2} \right)^{x} \), you could generate a table like:

• \( x = -3, f(x) = 8 \)
• \( x = -2, f(x) = 4 \)
• \( x = -1, f(x) = 2 \)
• \( x = 0, f(x) = 1 \)
• \( x = 1, f(x) = 0.5 \)
• \( x = 2, f(x) = 0.25 \)
• \( x = 3, f(x) = 0.125 \)

Such a table demonstrates the exponential decay of the function.
As x-values increase, y-values decrease, indicating that this function reflects a rapid decline, characteristic of functions where the base is less than 1.

Graph Sketching

Sketching the graph of an exponential function provides a visual representation that deepens understanding. Once you have your table of values, plot each point on a coordinate plane to aid in sketching the graph curve.

For \( f(x) = \left( \frac{1}{2} \right)^{x} \), plot the points: \((-3, 8), (-2, 4), (-1, 2), (0, 1), (1, 0.5), (2, 0.25), (3, 0.125)\).
This plotting forms a guide for your sketch. While sketching, notice how the graph starts from higher y-values and descends as x increases, illustrating a decreasing exponential function.
Additionally, observe the horizontal asymptote as the x-values approach infinity. The line never quite reaches zero, echoing the mathematical property that \( \left( \frac{1}{2} \right)^{x} \) tends towards zero but never actually equals it. This practice will fortify your grasp of how exponential functions behave graphically.