### Transformation of Exponential Functions#### **Understanding the Transformation:**The graph below illustrates a transformation of the exponential function $ y = 2^x $.![Graph](link_to_image) (Note: Replace this with the actual link if available.)#### **Graph Analysis:**- The original function $ y = 2^x $ is an exponential function that passes through the point (0, 1) since any number raised to the zero power is 1.- In the given graph, instead of passing through (0, 1), the graph appears to be reflected and translated.#### **Identifying the Transformation:**- **Reflection:** The graph appears to be reflected about the y-axis, indicating that the exponent $ x $ has been negated.- **Translation:** The graph does not appear to be simply reflected but also appears to have been shifted downwards by 1 unit based on the alignment of the curve.#### **Determining the Equation:**Considering the reflection and vertical translation, the equation can be derived as follows:\[ y = -2^x - 1 \]#### **Additional Exercises:**Next, verify the points:- $ f(-1) $ should align with the point (1, 1) on the new graph under reflection and translation.- Ensure continuity by checking multiple points to confirm the exact nature of transformation.### **Interactive Component:**Write an equation for the graph above:\[ y = \underline{\hspace{2cm}} \][Preview]### **Conclusion:**Understanding transformations of functions, particularly exponential ones, is crucial in advanced algebra. By reflecting and shifting, we explore the dynamic nature of equations and their graphical representations.

Answered: The graph below shows a transformation…

Math

Calculus

The graph below shows a transformation of y = 2"

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

### Transformation of Exponential Functions

#### **Understanding the Transformation:**
The graph below illustrates a transformation of the exponential function $ y = 2^x $.

![Graph](link_to_image) (Note: Replace this with the actual link if available.)

#### **Graph Analysis:**
- The original function $ y = 2^x $ is an exponential function that passes through the point (0, 1) since any number raised to the zero power is 1.
- In the given graph, instead of passing through (0, 1), the graph appears to be reflected and translated.

#### **Identifying the Transformation:**
- **Reflection:** The graph appears to be reflected about the y-axis, indicating that the exponent $ x $ has been negated.
- **Translation:** The graph does not appear to be simply reflected but also appears to have been shifted downwards by 1 unit based on the alignment of the curve.

#### **Determining the Equation:**
Considering the reflection and vertical translation, the equation can be derived as follows:
\[ y = -2^x - 1 \]

#### **Additional Exercises:**
Next, verify the points:
- $ f(-1) $ should align with the point (1, 1) on the new graph under reflection and translation.
- Ensure continuity by checking multiple points to confirm the exact nature of transformation.

### **Interactive Component:**
Write an equation for the graph above:
\[ y = \underline{\hspace{2cm}} \]
[Preview]

### **Conclusion:**
Understanding transformations of functions, particularly exponential ones, is crucial in advanced algebra. By reflecting and shifting, we explore the dynamic nature of equations and their graphical representations.

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