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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![**Mathematical Graphing Exercise**
**Objective:** Graph the following function and understand its transformations:
\[ f(\theta) = -3 \cot \left( \theta + \frac{3\pi}{4} \right) - 1 \]
**Domain:** \( -\pi \leq \theta \leq \pi \)
### Instructions:
1. **Equation Analysis:**
- Recognize the basic function: \( \cot(\theta) \).
- Identify the horizontal shift: \( \theta + \frac{3\pi}{4} \) (shifts left by \( \frac{3\pi}{4} \)).
- Observe the vertical stretch and reflection: \( -3 \cot \) (reflects over the x-axis and stretches vertically by a factor of 3).
- Note the vertical shift: \( -1 \) (shifts down by 1 unit).
2. **Graphing Steps:**
- Start by sketching the standard \( \cot(\theta) \) function.
- Apply the horizontal shift to the left by \( \frac{3\pi}{4} \).
- Reflect the graph vertically and apply the vertical stretch.
- Shift the entire graph downward by 1 unit.
3. **Character Limit:**
- Ensure that your explanation does not exceed 2000 characters.
### Graph Description:
- Unfortunately, the image does not contain a completed graph or diagram, so students need to graph the function on graph paper or using a graphing tool following the steps outlined above.
### Note:
- Make sure to label important points, asymptotes, and intercepts accurately.
- Pay attention to the range of \( \theta \) given by \( -\pi \leq \theta \leq \pi \).
Happy graphing!](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4c67c4d0-217c-4e3e-983c-7902447cdce8%2Ff8165711-c2b0-47e8-b35a-2f779b76fbdd%2F8tgwy5q_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Mathematical Graphing Exercise**
**Objective:** Graph the following function and understand its transformations:
\[ f(\theta) = -3 \cot \left( \theta + \frac{3\pi}{4} \right) - 1 \]
**Domain:** \( -\pi \leq \theta \leq \pi \)
### Instructions:
1. **Equation Analysis:**
- Recognize the basic function: \( \cot(\theta) \).
- Identify the horizontal shift: \( \theta + \frac{3\pi}{4} \) (shifts left by \( \frac{3\pi}{4} \)).
- Observe the vertical stretch and reflection: \( -3 \cot \) (reflects over the x-axis and stretches vertically by a factor of 3).
- Note the vertical shift: \( -1 \) (shifts down by 1 unit).
2. **Graphing Steps:**
- Start by sketching the standard \( \cot(\theta) \) function.
- Apply the horizontal shift to the left by \( \frac{3\pi}{4} \).
- Reflect the graph vertically and apply the vertical stretch.
- Shift the entire graph downward by 1 unit.
3. **Character Limit:**
- Ensure that your explanation does not exceed 2000 characters.
### Graph Description:
- Unfortunately, the image does not contain a completed graph or diagram, so students need to graph the function on graph paper or using a graphing tool following the steps outlined above.
### Note:
- Make sure to label important points, asymptotes, and intercepts accurately.
- Pay attention to the range of \( \theta \) given by \( -\pi \leq \theta \leq \pi \).
Happy graphing!
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