Problem 36 - Intermediate Algebra

Question

For Problems \(35-52\), graph each exponential function. $$ f(x)=3^{x} $$

Step-by-Step Solution

Answer

Plot points for x-values, graph dotted asymptote at y=0, and draw a curve through points.

Step 1: Identify the Function Type

The function given is an exponential function because it has the form \( f(x) = a^x \), where \( a > 0 \). In this case, \( f(x) = 3^x \) is an exponential function with base 3.

Step 2: Plot Key Points

To graph the exponential function, start by finding and plotting some key points. Choose a few values for \( x \) and calculate \( f(x) \):- For \( x = -2 \), \( f(-2) = 3^{-2} = \frac{1}{9} \).- For \( x = -1 \), \( f(-1) = 3^{-1} = \frac{1}{3} \).- For \( x = 0 \), \( f(0) = 3^0 = 1 \).- For \( x = 1 \), \( f(1) = 3^1 = 3 \).- For \( x = 2 \), \( f(2) = 3^2 = 9 \).

Step 3: Plot the Asymptote

Identify the horizontal asymptote of the exponential function. For any exponential function \( a^x \) where \( a > 0 \), the horizontal asymptote is \( y = 0 \). Graph a dashed line to represent the asymptote along the x-axis.

Step 4: Draw the Graph

Using the key points plotted, draw a smooth curve that passes through these points. The curve will start near the horizontal asymptote on the left and increase rapidly to the right as \( x \) becomes larger.

Step 5: Review the Characteristics of the Graph

Remember that the function \( 3^x \) is always positive for all real numbers \( x \), approaches zero as \( x \) approaches negative infinity, and increases without bound as \( x \) approaches positive infinity.

Key Concepts

Graphing

Graphing an exponential function, like \( f(x) = 3^x \), involves plotting points and drawing a smooth curve through them. This function is exponential because it follows the form \( f(x) = a^x \) with \( a > 0 \). Here, the base \( a \) is 3. To graph this, select key values of \( x \) to find corresponding \( f(x) \) values. Begin by choosing a few simple integers for \( x \), such as -2, -1, 0, 1, and 2.

• For \( x = -2 \), \( f(x) = \frac{1}{9} \).
• For \( x = -1 \), \( f(x) = \frac{1}{3} \).
• For \( x = 0 \), \( f(x) = 1 \).
• For \( x = 1 \), \( f(x) = 3 \).
• For \( x = 2 \), \( f(x) = 9 \).

With these points identified, plot them on the coordinate plane. Connect these points with a smooth curve to represent \( f(x) = 3^x \).
The curve will rise sharply from left to right, reflecting the rapid increase typical in exponential functions.

Horizontal Asymptote

A key feature to recognize when graphing exponential functions is the horizontal asymptote. This line indicates the behavior of the graph as \( x \) moves toward negative infinity. For exponential functions like \( f(x) = 3^x \), a horizontal asymptote will often be at \( y = 0 \).
This means as \( x \) becomes very negative, \( f(x) \) approaches but never reaches 0. Graph this asymptote as a dashed line along the x-axis. Remember, while the graph gets closer and closer to this line, it will not touch or cross it.
This horizontal asymptote helps visualize how the exponential function behaves, providing a boundary that suggests its shape and limits.

Function Characteristics

Understanding the characteristics of an exponential function such as \( f(x) = 3^x \) is crucial to fully grasp its behavior.
Here are some important points:

• The function is always positive. This means \( f(x) > 0 \) for all real values of \( x \).
• As \( x \) approaches negative infinity, \( f(x) \) approaches 0, closely following the horizontal asymptote \( y = 0 \).
• The function increases without bound as \( x \) becomes larger. Essentially, \( f(x) \) grows exponentially, rapidly gaining in value as \( x \) increases.

These characteristics give the function its unique shape and dictate how it behaves across different values of \( x \). By understanding these traits, you'll better predict and graph these exponential functions accurately.