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

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

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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**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 \(
Transcribed Image Text:**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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