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Magazine Chapter 5: Problem 48
Sketch the graph of the function and check the graph with a graphing calculator. Describe how each graph can be obtained from the graph of a basic exponential function. $$f(x)=1-e^{-0.01 x}$$
Short Answer
Expert verified
Graph of \(f(x) = 1 - e^{-0.01x}\) is a reflected and shifted version of \(e^x \) that approaches 1 as \(x\) increases.
Step by step solution
01
Identify the basic exponential function
The function given is related to the basic exponential function. The basic function of interest is \(e^{-0.01x}\). Start by understanding the shape of \(e^x\). The graph is a curve that increases rapidly from left to right.
02
Transform the basic function
To graph \(e^{-0.01x}\), observe how the negative sign in the exponent and the multiplication factor \(0.01\) stretch and reflect the basic exponential curve \(e^x\). This graph will decay slowly as \ x \ increases.
03
Apply transformations
For \(f(x)=1-e^{-0.01x}\), the graph of \(e^{-0.01x}\) is subtracted from 1. This transformation shifts the graph upwards by 1 unit and then reflects it across the x-axis. The curve approaches 1 as \ x \ increases.
04
Sketch the graph
Now sketch the graph. It starts at \(f(0) = 0\), increases as \ x \ increases, and asymptotically approaches \(f(x) = 1 \). The function never actually reaches 1 but gets infinitely close to it.
05
Check with graphing calculator
Use a graphing calculator to input \(f(x) = 1 - e^{-0.01x}\), and compare it with your sketch. Verify that the calculator's graph matches your expectations: starts near 0 when \ x = 0 \ and approaches 1 as \ x \ grows larger.
06
Describe transformations from \(e^x\)
Starting from \(e^x\), apply these transformations: reflect across the y-axis, horizontally stretch by a factor of \(100\), reflect across the x-axis, and shift upward by 1 unit to obtain \(f(x) = 1 - e^{-0.01x}\).
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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 mathematical expressions in which a constant base is raised to a variable exponent. The basic exponential function is expressed as \( e^x \), where \( e \) is Euler's number, approximately equal to \( 2.718 \). A key characteristic of exponential functions is their rapid growth or decay, which depends on the sign of the exponent.
- When the exponent is positive, the function exhibits exponential growth.
- When the exponent is negative, the function exhibits exponential decay.
Graphing Transformations
When graphing exponential functions, transformations such as stretching, shifting, and reflecting are often applied to a basic exponential graph. Let's break down the transformations for the function \( f(x) = 1 - e^{-0.01x} \):
- Start with \( e^x \): An exponentially growing function.
- Reflect across the y-axis: The function becomes \( e^{-x} \), now decaying.
- Horizontal stretch: Modify the exponent to \( -0.01x \). The factor of \( 0.01 \) causes the function to decay more slowly.
- Reflect across the x-axis: This transforms the function into \( -e^{-0.01x} \).
- Shift upward by 1 unit: Finally, \( 1 - e^{-0.01x} \) moves the entire graph up by one unit.
Asymptotic Behavior
Asymptotic behavior describes how a function behaves as its input grows very large or very small. For exponential functions like \( f(x) = 1 - e^{-0.01x} \), understanding the asymptotic behavior is crucial:
The graph of this function approaches an asymptote at \( y = 1 \) as \( x \) increases.
Notice that no matter how large \( x \) gets, the value of \( f(x) \) never quite reaches \( 1 \). This is due to the properties of exponentials:
The graph of this function approaches an asymptote at \( y = 1 \) as \( x \) increases.
Notice that no matter how large \( x \) gets, the value of \( f(x) \) never quite reaches \( 1 \). This is due to the properties of exponentials:
- \( e^{-0.01x} \) gets smaller and smaller as \( x \) increases, approaching zero but never hitting it.
- Subsequently, \( 1 - e^{-0.01x} \) gets closer and closer to 1.
Graphing Calculators
Graphing calculators are valuable tools for visualizing complex functions and verifying hand-drawn graphs. When dealing with exponential functions and their transformations, you can use these tools to:
Understanding how to use a graphing calculator can deepen your understanding of transformations and large-scale behaviors, bridging the gap between hand-drawn plots and mathematical prediction.
- Input functions: Enter your equation into the graphing calculator.
- View the graph: Observe how the calculator plots the function, noting key points and asymptotes.
Understanding how to use a graphing calculator can deepen your understanding of transformations and large-scale behaviors, bridging the gap between hand-drawn plots and mathematical prediction.
