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Chapter 10: Problem 75

In the following exercises, graph each exponential function. $$ f(x)=(0.4)^{x} $$

Short Answer

Expert verified
Plot points (-2, 6.25), (-1, 2.5), (0, 1), (1, 0.4), and (2, 0.16) and connect them with a smooth curve.

Step by step solution

01

Identify Key Points

To graph the function, start by identifying key points. Choose a few values for \( x \) and calculate corresponding \( y \) values for \( f(x) = (0.4)^{x} \). For example, let \( x \) be -2, -1, 0, 1, and 2.
02

Calculate Y-values

Compute the values for the function at chosen points: \[ f(-2) = (0.4)^{-2} = \frac{1}{(0.4)^{2}} \ f(-1) = (0.4)^{-1} = \frac{1}{0.4} \ f(0) = (0.4)^{0} = 1 \ f(1) = (0.4)^{1} = 0.4 \ f(2) = (0.4)^{2} = 0.16 \]
03

Plot the Points

Using the points calculated, plot them on a coordinate grid. The points are (-2, 6.25), (-1, 2.5), (0, 1), (1, 0.4), and (2, 0.16).
04

Draw the Graph

Connect the plotted points with a smooth curve. The curve should approach the horizontal axis (x-axis) as \( x \) increases but will never touch it. The graph will also rise steeply for negative values of \( x \).
05

Analyze the Graph's Behavior

Note that the exponential function \( f(x)=(0.4)^{x} \) is decreasing since the base of the exponent (0.4) is between 0 and 1. The y-values will get closer to 0 as x increases.

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Key Concepts

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

Exponential Functions
Exponential functions are a type of mathematical function involving an exponent. The basic form of an exponential function is given by \( f(x) = a^{x} \), where \( a \) is a constant called the base and \( x \) is the variable. In this exercise, the base is 0.4.
An important trait of exponential functions is how they grow or decay:
  • If \( a > 1 \), the function experiences exponential growth.
  • If \( 0 < a < 1 \), there is exponential decay, as in our example with 0.4.
Understanding the behavior of exponential functions is crucial for graphing them accurately.
Plotting Points
To graph exponential functions, start by plotting points. This involves choosing specific values for \( x \) and computing the corresponding \( y \) values called \( f(x) \).
Follow these steps:
  • Choose a range of values for \( x \). In our case, we chose \( x = -2, -1, 0, 1, \) and \( 2 \).
  • Calculate the \( y \)-values using the given function. For \( f(x) = (0.4)^{x} \):
    \( f(-2) = (0.4)^{-2} = \frac{1}{(0.4)^{2}} = 6.25 \)
    \( f(-1) = (0.4)^{-1} = \frac{1}{0.4} = 2.5 \)
    \( f(0) = (0.4)^{0} = 1 \)
    \( f(1) = (0.4)^{1} = 0.4 \)
    \( f(2) = (0.4)^{2} = 0.16 \)
Now, you have key points: (-2, 6.25), (-1, 2.5), (0, 1), (1, 0.4), and (2, 0.16).
Graph Analysis
After plotting these points, you can start drawing the actual graph.
Some important aspects to keep in mind while doing graph analysis include:
  • Connecting the plotted points using a smooth curve. The curve should not be broken or jagged.
  • Observing how the graph behaves as \( x \) increases or decreases. For \( f(x) = (0.4)^{x} \), the graph approaches the x-axis but never touches it.
  • Noting how the graph rises steeply for negative values of \( x \) since exponential decay reverses for negative exponents.
By analyzing these traits, you can better understand the overall shape and behavior of your exponential graph.
Function Behavior
Understanding the behavior of exponential functions helps in visualizing and graphing them effectively.
For the function \( f(x) = (0.4)^{x} \), the following behaviors are observed:
  • The function is decreasing because the base 0.4 is between 0 and 1. This results in exponential decay.
  • As \( x \) increases, the \( y \)-values get closer to 0, but the function never actually touches the x-axis. This is known as an asymptote.
  • For negative values of \( x \), the function values increase steeply, reflecting the reciprocal relationship in the exponentiation.
By grasping these points about function behavior, you enhance your ability to graph and interpret exponential functions.

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