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Use the guidelines of this section to sketch the curve. x - 5

Use the guidelines of this section to sketch the curve. x - 5

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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**Guidelines for Sketching a Curve** The following checklist is intended as a guide to sketching a curve \( y = f(x) \) by hand. Not every item is relevant to every function. (For instance, a given curve might not have an asymptote or possess symmetry.) But the guidelines provide all the information you need to make a sketch that displays the most important aspects of the function. ### A. Domain It's often useful to start by determining the domain \( D \) of \( f \), that is, the set of values of \( x \) for which \( f(x) \) is defined. ### B. Intercepts The y-intercept is \( f(0) \) and this tells us where the curve intersects the y-axis. To find the x-intercepts, we set \( y = 0 \) and solve for \( x \). (You can omit this step if the equation is difficult to solve.) ### C. Symmetry 1. **Even Function**: If \( f(-x) = f(x) \) for all \( x \) in \( D \), then \( f \) is an even function and the curve is symmetric about the y-axis. This means you only need to reflect the known part of the curve about the y-axis to obtain the complete curve. Examples include \( y = x^2, y = x^4, y = |x|, y = \cos x \). 2. **Odd Function**: If \( f(-x) = -f(x) \) for all \( x \) in \( D \), then \( f \) is an odd function and the curve is symmetric about the origin. Examples include \( y = x, y = x^3, y = x^5, y = \sin x \). 3. **Periodic Function**: If \( f(x + p) = f(x) \) for all \( x \), the curve is periodic with the smallest such number \( p \) as the period. If you know the shape over an interval of length \( p \), you can replicate it. \( y = \sin x \) and \( y = \tan x \) are periodic functions. **Figure Explanation**: - **Figure 1(a)**: Depicts an even function with reflectional symmetry about the y-axis. - **Figure 1(b)**: Shows an odd
Transcribed Image Text:**Guidelines for Sketching a Curve** The following checklist is intended as a guide to sketching a curve \( y = f(x) \) by hand. Not every item is relevant to every function. (For instance, a given curve might not have an asymptote or possess symmetry.) But the guidelines provide all the information you need to make a sketch that displays the most important aspects of the function. ### A. Domain It's often useful to start by determining the domain \( D \) of \( f \), that is, the set of values of \( x \) for which \( f(x) \) is defined. ### B. Intercepts The y-intercept is \( f(0) \) and this tells us where the curve intersects the y-axis. To find the x-intercepts, we set \( y = 0 \) and solve for \( x \). (You can omit this step if the equation is difficult to solve.) ### C. Symmetry 1. **Even Function**: If \( f(-x) = f(x) \) for all \( x \) in \( D \), then \( f \) is an even function and the curve is symmetric about the y-axis. This means you only need to reflect the known part of the curve about the y-axis to obtain the complete curve. Examples include \( y = x^2, y = x^4, y = |x|, y = \cos x \). 2. **Odd Function**: If \( f(-x) = -f(x) \) for all \( x \) in \( D \), then \( f \) is an odd function and the curve is symmetric about the origin. Examples include \( y = x, y = x^3, y = x^5, y = \sin x \). 3. **Periodic Function**: If \( f(x + p) = f(x) \) for all \( x \), the curve is periodic with the smallest such number \( p \) as the period. If you know the shape over an interval of length \( p \), you can replicate it. \( y = \sin x \) and \( y = \tan x \) are periodic functions. **Figure Explanation**: - **Figure 1(a)**: Depicts an even function with reflectional symmetry about the y-axis. - **Figure 1(b)**: Shows an odd
### Sketching the Curve: \( y = \frac{x}{x - 5} \) This section illustrates different sketches of the graph for the function \( y = \frac{x}{x - 5} \). The function has a vertical asymptote where the denominator is zero, i.e., \( x = 5 \). #### Graph Descriptions: 1. **Top Left Graph:** - The curve approaches a vertical asymptote at \( x = 5 \). - For \( x < 5 \), the curve approaches the asymptote from the left and rises steeply after crossing the x-axis. - For \( x > 5 \), the curve slopes upwards to the right, starting from high positive y-values. 2. **Top Right Graph:** - The graph exhibits a vertical asymptote at \( x = 5 \). - For \( x < 5 \), the curve is located in the third quadrant and approaches the asymptote from negative infinity. - For \( x > 5 \), the curve descends towards the asymptote from above. 3. **Bottom Left Graph:** - Exhibits a vertical asymptote at \( x = 5 \). - For \( x < 5 \), the graph appears in the second quadrant, approaching the asymptote from positive values above. - For \( x > 5 \), the curve is in the fourth quadrant, descending steeply towards negative infinity. 4. **Bottom Right Graph:** - Shows a vertical asymptote at \( x = 5 \). - For \( x < 5 \), the function is in the fourth quadrant reaching towards positive values. - For \( x > 5 \), the graph is positioned in the first quadrant, moving upward and away from the asymptote. Each graph represents a potential way the function could behave given its asymptote and the mathematical rules governing rational functions. The task is to determine which sketch correctly represents \( y = \frac{x}{x - 5} \).
Transcribed Image Text:### Sketching the Curve: \( y = \frac{x}{x - 5} \) This section illustrates different sketches of the graph for the function \( y = \frac{x}{x - 5} \). The function has a vertical asymptote where the denominator is zero, i.e., \( x = 5 \). #### Graph Descriptions: 1. **Top Left Graph:** - The curve approaches a vertical asymptote at \( x = 5 \). - For \( x < 5 \), the curve approaches the asymptote from the left and rises steeply after crossing the x-axis. - For \( x > 5 \), the curve slopes upwards to the right, starting from high positive y-values. 2. **Top Right Graph:** - The graph exhibits a vertical asymptote at \( x = 5 \). - For \( x < 5 \), the curve is located in the third quadrant and approaches the asymptote from negative infinity. - For \( x > 5 \), the curve descends towards the asymptote from above. 3. **Bottom Left Graph:** - Exhibits a vertical asymptote at \( x = 5 \). - For \( x < 5 \), the graph appears in the second quadrant, approaching the asymptote from positive values above. - For \( x > 5 \), the curve is in the fourth quadrant, descending steeply towards negative infinity. 4. **Bottom Right Graph:** - Shows a vertical asymptote at \( x = 5 \). - For \( x < 5 \), the function is in the fourth quadrant reaching towards positive values. - For \( x > 5 \), the graph is positioned in the first quadrant, moving upward and away from the asymptote. Each graph represents a potential way the function could behave given its asymptote and the mathematical rules governing rational functions. The task is to determine which sketch correctly represents \( y = \frac{x}{x - 5} \).
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