Problem 32 - Algebra for College Students

Question

Graph each of the exponential functions. $$ f(x)=\left(\frac{2}{3}\right)^{x} $$

Step-by-Step Solution

Answer

Graph \( f(x) = \left(\frac{2}{3}\right)^{x} \) with points showing exponential decay and a horizontal asymptote at \( y = 0 \).

Step 1: Understanding the Function

The given function is an exponential function of the form \( f(x) = b^x \), where \( b = \frac{2}{3} \). In this function, the base \( b \) is between 0 and 1, indicating that the graph will show exponential decay.

Step 2: Identify the Key Characteristics

Since \( b = \frac{2}{3} < 1 \), the graph will decrease as \( x \) increases. The y-intercept is at \( f(0) = 1 \), and there is an asymptote at \( y = 0 \), where the function approaches but never touches the x-axis.

Step 3: Create a Table of Values

To graph the function, calculate some key points: for example, \( f(-2), f(-1), f(0), f(1), \) and \( f(2) \). This will provide a sufficient set of points for sketching the graph.\[\begin{align*}f(-2) &= \left(\frac{2}{3}\right)^{-2} = \frac{9}{4}, \f(-1) &= \left(\frac{2}{3}\right)^{-1} = \frac{3}{2}, \f(0) &= \left(\frac{2}{3}\right)^{0} = 1, \f(1) &= \left(\frac{2}{3}\right)^{1} = \frac{2}{3}, \f(2) &= \left(\frac{2}{3}\right)^{2} = \frac{4}{9}.\end{align*}\]

Step 4: Plot the Points

Plot the calculated points on a graph: \((-2, \frac{9}{4})\), \((-1, \frac{3}{2})\), \((0, 1)\), \((1, \frac{2}{3})\), and \((2, \frac{4}{9})\). These points illustrate the exponential decay.

Step 5: Draw and Label the Graph

Connect the points plotted in the previous step with a smooth curve, ensuring it approaches the x-axis but does not touch or cross it. Label the y-intercept at \((0, 1)\) and highlight the horizontal asymptote at \( y = 0 \).

Key Concepts

Graphing Exponential Functions

To graph an exponential function such as \( f(x) = \left(\frac{2}{3}\right)^{x} \), start by understanding its basic structure. These functions have the form \( f(x) = b^x \), where \( b \) is a constant base. For graphing, create a table of key values by substituting a range of \( x \) values, like -2, -1, 0, 1, and 2. This will give you points on the graph. As you plot these points, you'll notice that an exponential function typically forms a smooth curve.

Remember to:

• Identify the y-intercept where \( x = 0 \).
• Identify any asymptotes to determine where the graph approaches but does not cross the axes.
• Connect the plotted points with a smooth curve for an accurate representation of the function.

Exponential Decay

Exponential decay occurs when the base \( b \) of the exponential function is between 0 and 1. This causes the function's value to decrease as \( x \) increases. In \( f(x) = \left(\frac{2}{3}\right)^{x} \), the base \( \frac{2}{3} \) indicates exponential decay. As you graph it by plotting points, you will see the curve descending rapidly.

• This tells us that smaller positive increments for \( x \) reduce \( f(x) \) significantly.
• The graph maintains its decreasing trend, showing how exponential decay models scenarios where quantities reduce over time.

By visualizing these characteristics on a graph, students can better comprehend how values evolve over time with exponential decay.

Asymptotes

In the realm of graphing, asymptotes are lines that the graph of a function approaches but never actually reaches. For the function \( f(x) = \left( \frac{2}{3} \right)^{x} \), there is a horizontal asymptote at \( y = 0 \).

This means that no matter how large \( x \) becomes, the value of \( f(x) \) edges closer to zero without really touching the x-axis. The horizontal line \( y = 0 \) plays a crucial role in defining the behavior of the exponential function over an extended range of \( x \) values.

• Understanding asymptotes helps in sketching graphs accurately.
• It provides insight into the end behavior of exponential functions.

Exponential Function Properties

Exponential functions have unique properties that distinguish them from other types of functions. Recognizing these properties helps in understanding their behavior and applications.

Key properties include:

• Base affects growth or decay: If \( b > 1 \), the function shows exponential growth. If \( 0 < b < 1 \), it shows decay.
• Continuous and smooth curve: The graph is not defined by any breaks.
• Y-intercept at 1: In \( f(x) = b^x \), the y-intercept occurs at \( f(0) = 1 \) because any non-zero number to the power of 0 equals 1.
• Asymptotic behavior: Exponential functions often have horizontal asymptotes that provide insight into limit behavior as \( x \to \infty \) or \( x \to -\infty \).
• Rate of change: Exponential functions increase or decrease more rapidly than linear or polynomial functions.

These properties underscore why exponential functions are crucial in modeling real-world phenomena like population growth or radioactive decay.