**Graphing Rational Functions Using Transformations**8) **Task:** Graph the rational function $ R(x) = \frac{2}{(x-3)^2} $ using transformations. Determine the domain, range, asymptotes, and intercepts.**Explanation and Steps:**1. **Identify the Basic Function:** The function given is derived from the basic rational function $ f(x) = \frac{1}{x^2} $.2. **Transformations:** - **Horizontal Shift:** The function is horizontally shifted 3 units to the right, resulting in the transformation $ \frac{1}{(x-3)^2} $. - **Vertical Stretch:** The function is vertically stretched by a factor of 2, giving $ R(x) = \frac{2}{(x-3)^2} $.3. **Domain:** - The function is undefined at $ x = 3 $ due to division by zero in the denominator. - **Domain:** $ x \in (-\infty, 3) \cup (3, \infty) $4. **Range:** - Since $\frac{2}{(x-3)^2}$ is always positive, the range is all positive values. - **Range:** $ y \in (0, \infty) $5. **Asymptotes:** - **Vertical Asymptote:** At $ x = 3 $, the function is undefined, making it a vertical asymptote. - **Horizontal Asymptote:** As $ x \to \pm\infty $, $ R(x) \to 0 $, making $ y = 0 $ a horizontal asymptote.6. **Intercepts:** - **X-Intercept:** No x-intercept, as the function does not cross the x-axis. - **Y-Intercept:** Plugging in $ x = 0 $ provides the y-intercept $ R(0) = \frac{2}{(0-3)^2} = \frac{2}{9} $.7. **Graph Description:** - The graph shifts right to $ x = 3 $ and opens upwards. - It approaches but never touches the x-axis (horizontal asymptote) and has a vertical asymptote at \(

We will use transformations on f(x)=1x2 to sketch the rational function 2(x-3)2. So, first we will…

Answered: 8) Graph the rational function R(x) =…

8) Graph the rational function R(x) = using transformations. Determine the (x-3)2 domain, range, asymptotes, and intercepts.

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

**Graphing Rational Functions Using Transformations**

8) **Task:** Graph the rational function $ R(x) = \frac{2}{(x-3)^2} $ using transformations. Determine the domain, range, asymptotes, and intercepts.

**Explanation and Steps:**

1. **Identify the Basic Function:**
The function given is derived from the basic rational function $ f(x) = \frac{1}{x^2} $.

2. **Transformations:**
- **Horizontal Shift:** The function is horizontally shifted 3 units to the right, resulting in the transformation $ \frac{1}{(x-3)^2} $.
- **Vertical Stretch:** The function is vertically stretched by a factor of 2, giving $ R(x) = \frac{2}{(x-3)^2} $.

3. **Domain:**
- The function is undefined at $ x = 3 $ due to division by zero in the denominator.
- **Domain:** $ x \in (-\infty, 3) \cup (3, \infty) $

4. **Range:**
- Since $\frac{2}{(x-3)^2}$ is always positive, the range is all positive values.
- **Range:** $ y \in (0, \infty) $

5. **Asymptotes:**
- **Vertical Asymptote:** At $ x = 3 $, the function is undefined, making it a vertical asymptote.
- **Horizontal Asymptote:** As $ x \to \pm\infty $, $ R(x) \to 0 $, making $ y = 0 $ a horizontal asymptote.

6. **Intercepts:**
- **X-Intercept:** No x-intercept, as the function does not cross the x-axis.
- **Y-Intercept:** Plugging in $ x = 0 $ provides the y-intercept $ R(0) = \frac{2}{(0-3)^2} = \frac{2}{9} $.

7. **Graph Description:**
- The graph shifts right to $ x = 3 $ and opens upwards.
- It approaches but never touches the x-axis (horizontal asymptote) and has a vertical asymptote at \(

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