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Magazine Chapter 1: Problem 9
Sketch the shifted exponential curves. $$y=1-e^{x} \text { and } y=1-e^{-x}$$
Short Answer
Expert verified
The graphs of \( y=1-e^{x} \) and \( y=1-e^{-x} \) are shifts of basic exponential curves, reflected across \( y=1 \).
Step by step solution
01
Understand the Basic Exponential Graphs
The basic exponential functions are given by \( y = e^{-x} \) and \( y = e^{x} \). These functions represent an exponential decay and exponential growth, respectively. For \( y = e^{-x} \), the graph decreases as \( x \) increases. For \( y = e^{x} \), the graph increases as \( x \) increases.
02
Apply the Transformations
For both functions, a vertical shift transformation is applied. The transformation for the given functions is: adding 1 to the original exponential functions: \( y = 1 - e^{x} \) and \( y = 1 - e^{-x} \). This transformation will shift the basic graphs upwards by 1 unit.
03
Sketch the First Function: y = 1 - e^x
First, consider the graph of \( y = e^x \). This curve crosses the y-axis at y = 1 and rises steeply as \( x \) increases. For \( y = 1 - e^x \), reflect the graph across the line \( y = 1 \). The curve now intercepts the y-axis at \( y = 0 \) and decreases, approaching \( y = 1 \) from below as \( x \) goes to negative infinity.
04
Sketch the Second Function: y = 1 - e^{-x}
Consider the graph of \( y = e^{-x} \). This curve crosses the y-axis at y = 1 and decreases as \( x \) increases. For \( y = 1 - e^{-x} \), reflect the graph across \( y = 1 \). The curve intercepts the y-axis at \( y = 0 \) and increases, approaching \( y = 1 \) from below as \( x \) goes to positive infinity.
05
Verify Important Points and End Behavior
Check key points like intercepts: For both functions, y-intercepts are \((0, 0)\). As \( x \) increases, \( y = 1 - e^x \) heads towards -鈭�, while \( y = 1 - e^{-x} \) approaches 1. In the other direction, as \( x \) decreases, \( y = 1 - e^x \) approaches 1 and \( y = 1 - e^{-x} \) heads towards -鈭�.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Exponential Growth
Exponential growth is a fundamental concept in mathematics and nature, characterizing processes that increase over time. It's depicted by the equation \( y = e^x \), where \( e \) is the base of the natural logarithm, approximately equal to 2.71828. This equation generates a curve that starts low and rises steeper as \( x \) increases.
The key features of exponential growth include:
The key features of exponential growth include:
- Rapid Increase: As the value of \( x \) goes up, the function \( y = e^x \) grows extremely fast.
- Y-intercept: It crosses the y-axis at \( y = 1 \).
- Infinite Growth: As \( x \) approaches infinity, the function tends towards infinity.
Exponential Decay
Exponential decay is where values decrease over time, represented by \( y = e^{-x} \). In this form, the function starts high and decreases as \( x \) increases, reflecting many natural processes like cooling or radioactive decay.
Key characteristics include:
Key characteristics include:
- Rapid Decrease: The function drops quickly as \( x \) increases.
- Y-intercept: Initially, it crosses the y-axis at \( y = 1 \).
- Approaches Zero: As \( x \) goes to infinity, \( y = e^{-x} \) gets closer to zero but never truly reaches it.
Graph Transformations
Graph transformations alter the appearance of a graph based on certain modifications. Two common transformations include vertical shifts and reflections.
For the functions \( y = 1 - e^x \) and \( y = 1 - e^{-x} \), vertical shifts play a crucial role. A transformation adds or subtracts a constant from the function. Here, adding 1 moves the entire graph up by 1 unit. Additionally, the entire function is reflected around \( y = 1 \), changing the orientation of exponential growth to decay, and vice versa.
By understanding graph transformations, students can easily modify and predict how the graph of a function will look post-transformation.
For the functions \( y = 1 - e^x \) and \( y = 1 - e^{-x} \), vertical shifts play a crucial role. A transformation adds or subtracts a constant from the function. Here, adding 1 moves the entire graph up by 1 unit. Additionally, the entire function is reflected around \( y = 1 \), changing the orientation of exponential growth to decay, and vice versa.
By understanding graph transformations, students can easily modify and predict how the graph of a function will look post-transformation.
Vertical Shift
A vertical shift involves moving a graph up or down along the y-axis without altering its shape. This shift is determined by adding or subtracting a constant to the function.
In our exercise, the transformation of adding 1 to the exponential functions in \( y = 1 - e^x \) and \( y = 1 - e^{-x} \) results in their upward shift by one unit.
In our exercise, the transformation of adding 1 to the exponential functions in \( y = 1 - e^x \) and \( y = 1 - e^{-x} \) results in their upward shift by one unit.
- The point where the graph would cut the y-axis decreases to \( y = 0 \).
- The maximum value now approaches \( y = 1 \) instead of \( \,楼 \) in exponential growth, showing an interesting dynamic.
