Problem 18 - Introductory and Intermediate Algebra for College Students 4th

Question

Graph each function by making a table of coordinates. If applicable, use a graphing unility to confirm your hand-drawn graph. $$f(x)=5^{x}$$

Step-by-Step Solution

Answer

After creating a table of coordinates with the given function \(f(x) = 5^x\) and plotting these points, the function's graph quickly rises from near the x-axis for negative x-values and then climbs steeply for positive x-values, confirming the nature of exponential functions.

Step 1: Constructing a Table of Coordinates

Choose several values of \(x\) to substitute into the function \(f(x) = 5^x\). For accuracy, let's use \(x = -2\), \(x = -1\), \(x = 0\), \(x = 1\), and \(x = 2\). This gives the following table:\n\n| \(x\) | \(f(x)\) |\n|---|---|\n| -2 | 0.04 |\n| -1 | 0.2 |\n| 0 | 1 |\n| 1 | 5 |\n| 2 | 25 |

Step 2: Plotting the Points

Using the calculated values, plot the points onto the graph. The points will be (-2, 0.04), (-1, 0.2), (0, 1), (1, 5), and (2, 25).

Step 3: Drawing the Curve

After the points are plotted, join them with a smooth curve. The curve should rise rapidly for positive \(x\)-values and approach zero for negative \(x\)-values.

Step 4: Confirm with a Graphing Utility

If available, input \(f(x) = 5^x\) into a graphing utility to confirm that it matches the hand-drawn graph.

Key Concepts

Constructing a Table of Coordinates

When graphing exponential functions such as \( f(x) = 5^x \), constructing a table of coordinates is a practical first step. This process involves choosing input values for \( x \), plugging them into the function, and recording the resulting output values or function values, \( f(x) \).

For instance, you might start by selecting simple values of \( x \), like -2, -1, 0, 1, and 2. Plugging these into the function \( f(x) = 5^x \) you will find the corresponding outputs: 0.04, 0.2, 1, 5, and 25, respectively. Your table would look something like this, with each row containing an \( x \) value and its corresponding \( f(x) \). These pairs are the coordinates you’ll be plotting:

• (-2, 0.04)
• (-1, 0.2)
• (0, 1)
• (1, 5)
• (2, 25)

Ideally, select a range of values that show the growth trend of the exponential function, which is characterized by a rapid increase as \( x \) becomes positive.

Plotting Points on a Graph

After creating your table of coordinates, the next vital stage is transferring these points to the graph. Start with a properly scaled axis that can accommodate the ranges in your table. Plotting points is as straightforward as marking the spot on the graph that corresponds to each \( x \) and \( f(x) \) pair.

For example, the point (-2, 0.04) is found by moving two units to the left on the horizontal axis (for -2) and slightly above zero on the vertical axis (for 0.04). This process is repeated for all coordinates. It’s important to be precise, as accurate plotting is essential for a correct representation of the exponential function. Once all points are on the graph, you should see the beginnings of an exponential curve. Ensuring that the points are plotted correctly before connecting them is critical, since an error here could lead to misconceptions about the function's behavior.

Exponential Function Properties

Understanding the properties of exponential functions can greatly aid in their graphing. An exponential function like \( f(x) = 5^x \) demonstrates several key characteristics:

• Exponential Growth: For positive bases greater than one, the function increases rapidly as \( x \) increases.
• Horizontal Asymptote: The graph approaches, but never touches, the x-axis as \( x \) decreases. For \( f(x) = 5^x \), this asymptote is the line \( y = 0 \).
• Domain and Range: The domain of \( f(x) \) is all real numbers, while the range is limited to positive real numbers only.
• Intercept: The graph of any exponential function with a positive base will always pass through the point (0, 1), as any number raised to the zero power is 1.

Recognizing these properties assists in accurately graphing the function and in predicting its behavior without plotting every single point. For instance, knowing about the horizontal asymptote guides you not to plot points where the function has a negative value, since it's impossible for this type of exponential function.