**Using Transformations to Graph Rational Functions**To graph the rational function $ r(x) = \frac{4x - 17}{x - 4} $ by transforming the graph of $ y = \frac{1}{x} $, follow these steps:### Graph Description**Top Left Graph:**- The graph of the function is shown in purple and blue. - It mirrors the shape of a typical hyperbola, which is the plot of $ y = \frac{1}{x} $.- There is a vertical asymptote at $ x = 4 $ and a horizontal asymptote at $ y = 4 $.**Top Right Graph:**- Again, the hyperbolic shape is evident.- Similar features are present with the vertical asymptote at $ x = 4 $ and a horizontal asymptote at $ y = 4 $.**Bottom Left Graph:**- This graph displays the consistent hyperbolic form.- The vertical line $ x = 4 $ indicates the vertical asymptote. - The horizontal asymptote remains at $ y = 4 $.**Bottom Right Graph:**- Demonstrates the same structural pattern as the previous graphs.- The vertical asymptote occurs at $ x = 4 $, while the horizontal asymptote is at $ y = 4 $.### Graph Analysis- **Vertical Asymptote:** Occurs at $ x = 4 $ due to the denominator approaching zero.- **Horizontal Asymptote:** Found at $ y = 4 $, determined by the leading coefficients of the polynomial.These transformations of the base graph $ y = \frac{1}{x} $ depict how changes in the function $ r(x) = \frac{4x - 17}{x - 4} $ affect its graphical representation. The image shows a webpage with four graphs depicting functions on Cartesian coordinates. Each graph includes a vertical asymptote (dashed blue line) and a horizontal asymptote (dashed blue line), with a red curve indicating the function's behavior near these asymptotes.- **Top Left Graph**: The function approaches a vertical asymptote at $x = 0$ and a horizontal asymptote at $y = 0$. The red curve is located in the third and first quadrants.- **Top Right Graph**: This function also has a vertical asymptote at $x = 0$ and a horizontal asymptote at $y = 0$. The red curve is located in the fourth and second quadrants.- **Bottom Left Graph**: Similar to the others, this graph has vertical and horizontal asymptotes at $x = 0$ and $y = 0$, respectively. The red curve is prominently in the first and third quadrants.- **Bottom Right Graph**: The function has the vertical asymptote at $x = 0$ and the horizontal asymptote at $y = 0$. The red curve appears in the second and fourth quadrants.Below the graphs, there are input fields for entering the domain and range of the functions using interval notation. A checkbox beside these fields indicates that answers can be submitted for evaluation. There is also a "Need Help?" section with a "Read It" button for additional assistance. Please use interval notation to enter your answers for the domain and range of these functions.

r(x)=4x-17x-4r(x)=4x-16-1x-4r(x)=4(x-4)-1x-4r(x)=4(x-4)(x-4)-1(x-4)r(x)=4-1(x-4)r(x)=4-f(x-4)r(x)=-f…

Use transformations of the graph of y = 1/x to graph the rational function, as in Example 2. 4х - 17 r(x) = x- 4 y y 10 10- 5. -15 -10 -5 10. 15 -15 -10 10 15 wwww -5 -10 -10F 10 10 -15 -10 -15 -10 10 15 10 15 MacBook Air

Use transformations of the graph of y = 1/x to graph the rational function, as in Example 2. 4х - 17 r(x) = x- 4 y y 10 10- 5. -15 -10 -5 10. 15 -15 -10 10 15 wwww -5 -10 -10F 10 10 -15 -10 -15 -10 10 15 10 15 MacBook Air

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

**Using Transformations to Graph Rational Functions**

To graph the rational function $ r(x) = \frac{4x - 17}{x - 4} $ by transforming the graph of $ y = \frac{1}{x} $, follow these steps:

### Graph Description

**Top Left Graph:**
- The graph of the function is shown in purple and blue.
- It mirrors the shape of a typical hyperbola, which is the plot of $ y = \frac{1}{x} $.
- There is a vertical asymptote at $ x = 4 $ and a horizontal asymptote at $ y = 4 $.

**Top Right Graph:**
- Again, the hyperbolic shape is evident.
- Similar features are present with the vertical asymptote at $ x = 4 $ and a horizontal asymptote at $ y = 4 $.

**Bottom Left Graph:**
- This graph displays the consistent hyperbolic form.
- The vertical line $ x = 4 $ indicates the vertical asymptote.
- The horizontal asymptote remains at $ y = 4 $.

**Bottom Right Graph:**
- Demonstrates the same structural pattern as the previous graphs.
- The vertical asymptote occurs at $ x = 4 $, while the horizontal asymptote is at $ y = 4 $.

### Graph Analysis
- **Vertical Asymptote:** Occurs at $ x = 4 $ due to the denominator approaching zero.
- **Horizontal Asymptote:** Found at $ y = 4 $, determined by the leading coefficients of the polynomial.

These transformations of the base graph $ y = \frac{1}{x} $ depict how changes in the function $ r(x) = \frac{4x - 17}{x - 4} $ affect its graphical representation.

The image shows a webpage with four graphs depicting functions on Cartesian coordinates. Each graph includes a vertical asymptote (dashed blue line) and a horizontal asymptote (dashed blue line), with a red curve indicating the function's behavior near these asymptotes.

- **Top Left Graph**: The function approaches a vertical asymptote at $x = 0$ and a horizontal asymptote at $y = 0$. The red curve is located in the third and first quadrants.

- **Top Right Graph**: This function also has a vertical asymptote at $x = 0$ and a horizontal asymptote at $y = 0$. The red curve is located in the fourth and second quadrants.

- **Bottom Left Graph**: Similar to the others, this graph has vertical and horizontal asymptotes at $x = 0$ and $y = 0$, respectively. The red curve is prominently in the first and third quadrants.

- **Bottom Right Graph**: The function has the vertical asymptote at $x = 0$ and the horizontal asymptote at $y = 0$. The red curve appears in the second and fourth quadrants.

Below the graphs, there are input fields for entering the domain and range of the functions using interval notation. A checkbox beside these fields indicates that answers can be submitted for evaluation. There is also a "Need Help?" section with a "Read It" button for additional assistance.

Please use interval notation to enter your answers for the domain and range of these functions.

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