Homework Help is Here – Start Your Trial Now!
Math
Calculus
What is the graph of the function? 글 +3 -1 f (x) = x-2

What is the graph of the function? 글 +3 -1 f (x) = x-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...
See similar textbooks
icon
Related questions
Question
**Educational Website Content** **Graphing Rational Functions** **Function:** \[ f(x) = -\frac{1}{x-2} + 3 \] **Description:** We are given the function \( f(x) = -\frac{1}{x-2} + 3 \) and asked to identify its graph among two options displayed above. This function represents a rational function with a vertical asymptote and a horizontal transformation. **Graphs Analysis:** 1. **First Graph (Left):** - **Axes Range:** The x-axis ranges from -7 to 5, and the y-axis ranges from -5 to 5. - **Curve Behavior:** The curve approaches a vertical line (asymptote) around \( x = -1 \) and runs off to positive and negative infinities. It curves over the x-axis, moving from top-left to bottom-right. - **Horizontal Shift and Vertical Shift:** This graph displays a vertical asymptote at \( x = -1 \) and seems to have a horizontal asymptote as it approaches \( y = 0 \). 2. **Second Graph (Right):** - **Axes Range:** The x-axis ranges from -5 to 7, and the y-axis ranges from -5 to 7. - **Curve Behavior:** The curve has a vertical asymptote at \( x = 2 \) and another horizontal asymptote as it approaches \( y = 3 \). The curve moves from the top-right and bottom-left quadrants, which is consistent with the transformation of rational functions. - **Horizontal Shift and Vertical Shift:** This graph accurately reflects the function's transformation with vertical shift upwards by 3 units. **Correct Graph:** The second graph on the right accurately represents the function \( f(x) = -\frac{1}{x-2} + 3 \). It captures the vertical asymptote at \( x = 2 \) and the horizontal asymptote at \( y = 3 \). This is consistent with the properties of the given transformed rational function.
Transcribed Image Text:**Educational Website Content** **Graphing Rational Functions** **Function:** \[ f(x) = -\frac{1}{x-2} + 3 \] **Description:** We are given the function \( f(x) = -\frac{1}{x-2} + 3 \) and asked to identify its graph among two options displayed above. This function represents a rational function with a vertical asymptote and a horizontal transformation. **Graphs Analysis:** 1. **First Graph (Left):** - **Axes Range:** The x-axis ranges from -7 to 5, and the y-axis ranges from -5 to 5. - **Curve Behavior:** The curve approaches a vertical line (asymptote) around \( x = -1 \) and runs off to positive and negative infinities. It curves over the x-axis, moving from top-left to bottom-right. - **Horizontal Shift and Vertical Shift:** This graph displays a vertical asymptote at \( x = -1 \) and seems to have a horizontal asymptote as it approaches \( y = 0 \). 2. **Second Graph (Right):** - **Axes Range:** The x-axis ranges from -5 to 7, and the y-axis ranges from -5 to 7. - **Curve Behavior:** The curve has a vertical asymptote at \( x = 2 \) and another horizontal asymptote as it approaches \( y = 3 \). The curve moves from the top-right and bottom-left quadrants, which is consistent with the transformation of rational functions. - **Horizontal Shift and Vertical Shift:** This graph accurately reflects the function's transformation with vertical shift upwards by 3 units. **Correct Graph:** The second graph on the right accurately represents the function \( f(x) = -\frac{1}{x-2} + 3 \). It captures the vertical asymptote at \( x = 2 \) and the horizontal asymptote at \( y = 3 \). This is consistent with the properties of the given transformed rational function.
### Analyzing Graphs of Rational Functions The image contains two graphs displaying rational functions, each characterized by specific asymptotic behaviors. #### Left Graph - **Axes:** The graph is plotted with `x` on the horizontal axis ranging from -5 to 7, and `y` on the vertical axis ranging from -5 to 7. - **Curve Description:** - The graph consists of two branches. - The branch in the first quadrant curves upward and to the right, indicating positive values of `y` as `x` increases. - The second branch, located in the third quadrant, decreases as it moves leftwards, indicating negative values of `y`. - **Asymptotes:** - The vertical asymptote is likely around `x = 3`, suggesting the function approaches infinity as it gets closer to this line. - A horizontal asymptote may exist around `y = 3`, revealing that as `x` approaches infinity, `y` approaches this value. #### Right Graph - **Axes:** The graph is plotted similarly, with `x` on the horizontal axis ranging from -7 to 5, and `y` on the vertical axis from -5 to 7. - **Curve Description:** - This graph also consists of two branches. - The first branch, in the second quadrant, curves upward, suggesting a steep increase as `x` approaches the vertical asymptote. - The second branch in the fourth quadrant decreases sharply, indicating negative `y` values as `x` approaches the vertical asymptote from the right. - **Asymptotes:** - The vertical asymptote is likely around `x = -2`, suggesting the function approaches infinity or negative infinity near this line. - A horizontal asymptote appears around `y = 2`, indicating that as `x` increases positively or negatively, `y` nears this value. ### Conclusion These graphs demonstrate classic behaviors of rational functions where vertical asymptotes occur at values excluded from their domain, and horizontal asymptotes represent long-term behavior as `x` approaches positive or negative infinity. Understanding these characteristics is essential in analyzing and interpreting rational functions.
Transcribed Image Text:### Analyzing Graphs of Rational Functions The image contains two graphs displaying rational functions, each characterized by specific asymptotic behaviors. #### Left Graph - **Axes:** The graph is plotted with `x` on the horizontal axis ranging from -5 to 7, and `y` on the vertical axis ranging from -5 to 7. - **Curve Description:** - The graph consists of two branches. - The branch in the first quadrant curves upward and to the right, indicating positive values of `y` as `x` increases. - The second branch, located in the third quadrant, decreases as it moves leftwards, indicating negative values of `y`. - **Asymptotes:** - The vertical asymptote is likely around `x = 3`, suggesting the function approaches infinity as it gets closer to this line. - A horizontal asymptote may exist around `y = 3`, revealing that as `x` approaches infinity, `y` approaches this value. #### Right Graph - **Axes:** The graph is plotted similarly, with `x` on the horizontal axis ranging from -7 to 5, and `y` on the vertical axis from -5 to 7. - **Curve Description:** - This graph also consists of two branches. - The first branch, in the second quadrant, curves upward, suggesting a steep increase as `x` approaches the vertical asymptote. - The second branch in the fourth quadrant decreases sharply, indicating negative `y` values as `x` approaches the vertical asymptote from the right. - **Asymptotes:** - The vertical asymptote is likely around `x = -2`, suggesting the function approaches infinity or negative infinity near this line. - A horizontal asymptote appears around `y = 2`, indicating that as `x` increases positively or negatively, `y` nears this value. ### Conclusion These graphs demonstrate classic behaviors of rational functions where vertical asymptotes occur at values excluded from their domain, and horizontal asymptotes represent long-term behavior as `x` approaches positive or negative infinity. Understanding these characteristics is essential in analyzing and interpreting rational functions.
Expert Solution
Step 1

the solution is given below

steps

Step by step

Solved in 3 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Precalculus
Precalculus
Calculus
ISBN:
9780135189405
Author:
Michael Sullivan
Publisher:
PEARSON
Calculus: Early Transcendental Functions
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS