Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 5: Problem 22
First tell what translations or reflections are required to obtain the graph of the given function from that of \(y=e^{x} .\) Then graph the given function along with \(y=e^{x}\) and check that the picture is consistent with what you've said. (a) \(y=1-e^{-x}\) (b) \(y=1-e^{-x+1}\)
Short Answer
Expert verified
(a) Reflect around y-axis, translate up by 1; (b) Reflect, translate up by 1, shift right by 1.
Step by step solution
01
Understanding the Base Function
The base function is \(y = e^x\), which is an exponential function with a standard growth curve that passes through (0,1) and approaches infinity as \(x\) increases.
02
Analyzing Function (a) - Reflection and Translation
The function \(y = 1 - e^{-x}\) involves reflecting \(e^x\) in the y-axis through the substitution of \(x\) with \(-x\) (resulting in \(e^{-x}\)), followed by a vertical translation upwards by 1 unit (resulting in \(1 - e^{-x}\)).
03
Graphing Function (a)
For function \(y = 1 - e^{-x}\):1. Graph \(y = e^x\). 2. Reflect about the y-axis to get \(y = e^{-x}\).3. Shift the entire graph up by 1 unit to obtain the final function. The resulting graph should start at y=1 when x=0 and approach y=1 asymptotically from below as x approaches infinity.
04
Analyzing Function (b) - Reflection, Translation, and Horizontal Shift
For \(y = 1 - e^{-x+1}\), apply the reflection and vertical translation as in part (a). Additionally, apply a horizontal shift to the right by 1 unit given by the change to \(e^{-x+1} = e^{-(x-1)}\).
05
Graphing Function (b)
For function \(y = 1 - e^{-x+1}\): 1. Graph \(y = e^x\). 2. Reflect to get \(y = e^{-x}\).3. Shift to the right by 1 unit to get \(y = e^{-(x-1)}\).4. Finally, shift upwards by 1 unit to obtain the graph of \(y = 1 - e^{-x+1}\).This graph will approach y=1 asymptotically from below, starting 1 unit to the right compared to the translation in part (a).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graph Transformations
Graph transformations are actions that change the position, size, or orientation of a graph while maintaining its overall shape. Understanding these transformations allows us to manipulate the graph of a basic function, like an exponential function, into more complex forms.
For example, the function for an exponential graph, such as \(y = e^x\), can undergo various transformations to represent different scenarios or data. These transformations can be categorized into several types:
For example, the function for an exponential graph, such as \(y = e^x\), can undergo various transformations to represent different scenarios or data. These transformations can be categorized into several types:
- Vertical Shifts: Moving the graph up or down without altering its horizontal position.
- Horizontal Shifts: Adjusting the graph left or right without affecting its vertical position.
- Reflections: Flipping the graph over a specific line, such as the y-axis, to achieve a mirror image.
- Stretching or Compressing: Altering the graph's size either vertically or horizontally, affecting its steepness or width.
Reflections in the y-axis
A reflection in the y-axis involves flipping the graph of a function horizontally, creating a mirror image. This transformation is achieved by replacing every instance of \(x\) in the function with \(-x\).
This change causes points on the graph to shift positions relative to the y-axis. In the context of exponential functions like \(y = e^x\), reflecting in the y-axis transforms it to \(y = e^{-x}\).
It's essential to note how this transformation affects the graph:
This change causes points on the graph to shift positions relative to the y-axis. In the context of exponential functions like \(y = e^x\), reflecting in the y-axis transforms it to \(y = e^{-x}\).
It's essential to note how this transformation affects the graph:
- Every point \((x, y)\) becomes \((-x, y)\), maintaining the same y-coordinate but altering the x-coordinate.
- The graph of \(y = e^{-x}\) will approach the x-axis asymptotically as \(x\) approaches positive infinity.
- This reflection does not change the vertical positioning of the graph, but it does reverse the direction of its horizontal spread.
Vertical Translation
Vertical translation refers to moving a graph up or down on a coordinate plane. This shift is achieved by adding or subtracting a constant to the entire function, affecting all y-values uniformly.
For instance, if we consider the function \(y = e^{-x}\) and apply a vertical translation by adding 1, the function becomes \(y = 1 - e^{-x}\).
The effects of this shift are:
For instance, if we consider the function \(y = e^{-x}\) and apply a vertical translation by adding 1, the function becomes \(y = 1 - e^{-x}\).
The effects of this shift are:
- The graph maintains its original shape and direction.
- Every point \((x, y)\) on the graph is moved to \((x, y+c)\), where \(c\) is the constant added.
- The entire graph shifts upwards by 1 unit, ensuring all previous asymptotic behaviors, such as approaching the x-axis, remain intact but now occur 1 unit higher along the y-axis.
Horizontal Shift
A horizontal shift moves the graph left or right on the coordinate plane, achieved by adjusting the input of the function. This is done by replacing \(x\) with \(x-h\), where \(h\) is the shift amount.
In the exercise, when changing \(y = 1 - e^{-x}\) to \(y = 1 - e^{-x+1}\), we effectively apply a horizontal shift of the graph 1 unit to the right.
Here鈥檚 how this transformation plays out:
In the exercise, when changing \(y = 1 - e^{-x}\) to \(y = 1 - e^{-x+1}\), we effectively apply a horizontal shift of the graph 1 unit to the right.
Here鈥檚 how this transformation plays out:
- The function \(e^{-x}\) becomes \(e^{-(x-1)}\), implying an input shift rightward by 1 unit.
- Each point on the original graph moves horizontally, changing \((x, y)\) to \((x+h, y)\).
- The starting point and asymptotic behavior are both moved rightward without any alteration of the vertical dynamics.
