Problem 42 For Problems $35-52$, graph ea... [FREE SOLUTION]

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Intermediate Algebra

Jerome E. Kaufmann, Karen L. Schwitters

$Math Studyset 91影视 Explanations$ Math

9 Edition

Chapter 11: Problem 42

For Problems $35-52$, graph each exponential function. $$ f(x)=\left(\frac{2}{3}\right)^{x} $$

Short Answer

Expert verified

The graph of $ f(x) = \left(\frac{2}{3}\right)^x $ is an exponential decay curve that decreases as $ x $ increases.

Step by step solution

Identify the Type of Function

The function is an exponential function of the form $ f(x) = a^x $. In this case, $ a $ is $ \frac{2}{3} $. Since $ a < 1 $, the function represents exponential decay.

Create a Table of Values

Choose a few values for $ x $ to calculate corresponding $ f(x) $ values. For example:- $ x = -2 $: $ f(-2) = \left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^{2} = \frac{9}{4}$- $ x = -1 $: $ f(-1) = \left(\frac{3}{2}\right)^1 = \frac{3}{2} $- $ x = 0 $: $ f(0) = \left(\frac{2}{3}\right)^0 = 1 $- $ x = 1 $: $ f(1) = \left(\frac{2}{3}\right)^1 = \frac{2}{3} $- $ x = 2 $: $ f(2) = \left(\frac{2}{3}\right)^2 = \frac{4}{9} $

Plot the Points

Using the table of values, plot the points $(-2, \frac{9}{4}), (-1, \frac{3}{2}), (0, 1), (1, \frac{2}{3}), (2, \frac{4}{9})$ on a coordinate plane. These points represent the function's behavior.

Draw the Graph

Connect the plotted points with a smooth curve. As $ x $ approaches negative infinity, the graph approaches zero, indicating the function decreases as $ x $ increases. The curve should show a decreasing trend from left to right.

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Key Concepts

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

graphing exponential functions

Graphing exponential functions involves plotting two variables, typically denoted as $ x $ and $ f(x) $. The function $ f(x) = \left( \frac{2}{3} \right)^x $ is an exponential function, where the exponent $ x $ determines the value of the function.
To graph such a function, a step-by-step method is followed:

Choose a set of $ x $ values to generate corresponding $ f(x) $ values, forming a table of values (more on this later).
Use these computed points to plot them on a coordinate plane.
Connect these points smoothly to represent the trend accurately.

Exponential functions are not straight lines; they curve, showing either exponential growth or decay. The base of the exponential, in this case $ \frac{2}{3} $, plays a critical role in determining the type of curve.

exponential decay

Exponential decay occurs in exponential functions where the base of the exponent is between 0 and 1, such as $ \frac{2}{3} $. This results in the function output decreasing as $ x $ increases.
In the specific function $ f(x) = \left( \frac{2}{3} \right)^x $, the base is less than one, indicating decay:

This means that as time (or $ x $) increases, the value of $ f(x) $ gets closer and closer to zero.
It represents processes such as radioactive decay or depreciation in value over time.

Understanding exponential decay is crucial in many scientific and financial contexts, helping predict trends and behaviors of various systems.

coordinate plane

The coordinate plane, essential for graphing functions, consists of two axes: the horizontal $ x $-axis and the vertical $ y $-axis. Points on this plane have coordinates $(x, y)$, where $ y $ corresponds to $ f(x) $.
For an exponential function like $ f(x) = \left( \frac{2}{3} \right)^x $, plotting it involves identifying specific points using a table of values:

Select points calculated from the function, such as $(-2, \frac{9}{4})$, and plot each one on the plane.
The plane allows for a visual representation of how $ f(x) $ changes as $ x $ changes.

The coordinate plane is a powerful tool in visualizing and interpreting mathematical functions.

table of values

A table of values is a fundamental tool in graphing functions. It helps establish the relationship between $ x $ and $ f(x) $ by calculating and listing several points:

Choose $ x $ values that adequately capture the function's behavior, both negative and positive.
Calculate corresponding $ f(x) $ values for each $ x $.
List the results as pairs, such as $(x, f(x))$, that can be used for plotting.

In the exercise, the table included values like $ x = -2 $, giving $ f(-2)=\frac{9}{4} $, which helps in the accurate drawing of the curve on a coordinate plane. Creating a table of values ensures that the function's path is well-represented, leading to an accurate graph.

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91影视

For Problems \(35-52\), graph each exponential function. $$ f(x)=\left(\frac{2}{3}\right)^{x} $$

Short Answer

Step by step solution

Identify the Type of Function

Create a Table of Values

Plot the Points

Draw the Graph

Key Concepts

graphing exponential functions

exponential decay

coordinate plane

table of values

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