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1. Graph with all parts (in other words, list all transformations, show all important parts of graph, domain and range, etc.) y = 5 – log3 (z +2)

1. Graph with all parts (in other words, list all transformations, show all important parts of graph, domain and range, etc.) y = 5 – log3 (z +2)

Calculus: Early Transcendentals
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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**1. Graph with all parts** (in other words, list all transformations, show all important parts of graph, domain and range, etc.) \( y = 5 - \log_3 (x + 2) \) **Instructions for Display on Educational Website:** 1. **Function and Graph Overview:** - The function is \( y = 5 - \log_3 (x + 2) \). - This is a logarithmic function that has been transformed. 2. **Transformations:** - **Vertical Shift**: The function is shifted 5 units upwards because of the "+ 5" outside the logarithm. - **Horizontal Shift**: The graph is shifted 2 units to the left due to the "+ 2" inside the argument of the logarithm. - **Reflection**: The negative sign before the logarithm indicates a reflection across the x-axis. 3. **Important Parts of the Graph:** - **Vertical Asymptote**: At \( x = -2 \), because the logarithm becomes undefined at \( x + 2 = 0 \). - **Intercepts**: Calculate specific intercepts if necessary. 4. **Domain and Range:** - **Domain**: \( x > -2 \) (since \( x + 2 > 0 \) to keep the logarithm defined). - **Range**: All real numbers (since reflections and vertical shifts do not affect the range of a logarithmic function). 5. **Additional Notes:** - The base of the logarithm is 3, indicating moderate growth. - The graph passes through calculated points to validate the transformation steps. This explanation outlines how the transformations affect the graph and provides insight into analyzing similar logarithmic functions.
Transcribed Image Text:**1. Graph with all parts** (in other words, list all transformations, show all important parts of graph, domain and range, etc.) \( y = 5 - \log_3 (x + 2) \) **Instructions for Display on Educational Website:** 1. **Function and Graph Overview:** - The function is \( y = 5 - \log_3 (x + 2) \). - This is a logarithmic function that has been transformed. 2. **Transformations:** - **Vertical Shift**: The function is shifted 5 units upwards because of the "+ 5" outside the logarithm. - **Horizontal Shift**: The graph is shifted 2 units to the left due to the "+ 2" inside the argument of the logarithm. - **Reflection**: The negative sign before the logarithm indicates a reflection across the x-axis. 3. **Important Parts of the Graph:** - **Vertical Asymptote**: At \( x = -2 \), because the logarithm becomes undefined at \( x + 2 = 0 \). - **Intercepts**: Calculate specific intercepts if necessary. 4. **Domain and Range:** - **Domain**: \( x > -2 \) (since \( x + 2 > 0 \) to keep the logarithm defined). - **Range**: All real numbers (since reflections and vertical shifts do not affect the range of a logarithmic function). 5. **Additional Notes:** - The base of the logarithm is 3, indicating moderate growth. - The graph passes through calculated points to validate the transformation steps. This explanation outlines how the transformations affect the graph and provides insight into analyzing similar logarithmic functions.
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