Problem 21 Graph each exponential function.... [FREE SOLUTION]

Chapter 10: Problem 21

Graph each exponential function. $$ g(x)=\left(\frac{1}{3}\right)^{x} $$

Short Answer

Expert verified

The graph of $ g(x) = \left(\frac{1}{3}\right)^{x} $ is a decreasing exponential function with a horizontal asymptote at $ y = 0 $.

Step by step solution

Identify the Exponential Function

The given exponential function is $ g(x) = \left(\frac{1}{3}\right)^{x} $. Here, the base of the exponential function is $ \frac{1}{3} $.

Create a Table of Values

To graph the function, create a table of values for $ x $ and $ g(x) $. Choose some values for $ x $, both positive and negative, to see the behavior of the function.For example:$ x = -2, -1, 0, 1, 2 $Calculate $ g(x) $ for each $ x $:$ g(-2) = \left(\frac{1}{3}\right)^{-2} = 9 $$ g(-1) = \left(\frac{1}{3}\right)^{-1} = 3 $$ g(0) = \left(\frac{1}{3}\right)^{0} = 1 $$ g(1) = \left(\frac{1}{3}\right)^{1} = \frac{1}{3} $$ g(2) = \left(\frac{1}{3}\right)^{2} = \frac{1}{9} $

Plot the Points

Using the table of values, plot the points on a coordinate plane. The points are:($ -2, 9 $), ($ -1, 3 $), ($ 0, 1 $), ($ 1, \frac{1}{3} $), ($ 2, \frac{1}{9} $).

Draw the Graph

Connect the points with a smooth curve to complete the graph of the exponential function $ g(x) = \left(\frac{1}{3}\right)^{x} $. Note that as $ x $ increases, $ g(x) $ decreases toward 0. As $ x $ decreases, $ g(x) $ increases exponentially.

Analyze the Graph

The graph is a decreasing exponential function since the base $ \frac{1}{3} $ is between 0 and 1. The horizontal asymptote of the graph is the x-axis ($ y = 0 $).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Plotting Exponential Functions

Plotting exponential functions involves graphing the set of solutions to a function of the form $g(x) = a^x$, where $a$ is a constant. In this exercise, the function is $g(x)=\frac{1}{3}^x$. The process starts with creating a table of values.
For instance, you could choose $x = -2, -1, 0, 1, 2$. This allows you to see the behavior of the function with both positive and negative values of $x$.
Once you have your input values, calculate the corresponding $g(x)$ values:

When $x = -2$, $g(-2) = 9$
When $x = -1$, $g(-1) = 3$
When $x = 0$, $g(0) = 1$
When $x = 1$, $g(1) = \frac{1}{3}$
When $x = 2$, $g(2) = \frac{1}{9}$

Plot these points on a coordinate plane:

(-2, 9)
(-1, 3)
(0, 1)
(1, 1/3)
(2, 1/9)

After plotting, connect the points smoothly to reveal the graph.
The curve will exhibit a distinct shape typical of exponential functions.

Decreasing Exponential Functions

A decreasing exponential function is characterized by its base being between 0 and 1.
In our example, $\frac{1}{3}$ is the base. Since it's less than 1 and greater than 0, the function $g(x) = \frac{1}{3}^x$ decreases as $x$ increases. This tells us something important about the function's behavior.
Here are some key features to note:

As $x$ goes to positive infinity, $g(x)$ approaches 0.
For increasing values of $x$, $g(x)$ gets smaller and smaller.
Conversely, as $x$ becomes more negative, $g(x)$ grows larger.

This exponential decay is evident visually on the graph, where the slope of the curve declines steeply. Understanding this helps in predicting the values and behavior of the function over different ranges of $x$.

Horizontal Asymptote

In the context of exponential functions, the horizontal asymptote is a line that the graph approaches but never touches.
For the function $g(x) = \frac{1}{3}^x$, we observe that as $x$ increases indefinitely, the value of $g(x)$ gets closer and closer to 0 but never actually becomes 0.

This is why the line $y = 0$ (or the x-axis) is considered the horizontal asymptote. Some key points:

The function $g(x)$ will decrease infinitely but will never cross the x-axis.
The horizontal asymptote represents the lower boundary of the function's value.
This line acts as a reference point to understand how the function behaves as $x$ grows larger.

Knowing about the horizontal asymptote helps in understanding the long-term behavior of exponential functions, providing a clearer picture of how the function behaves at extreme values of $x$.

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91影视

Graph each exponential function. $$ g(x)=\left(\frac{1}{3}\right)^{x} $$

Short Answer

Step by step solution

Identify the Exponential Function

Create a Table of Values

Plot the Points

Draw the Graph

Analyze the Graph

Key Concepts

Plotting Exponential Functions

Decreasing Exponential Functions

Horizontal Asymptote

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