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Jse the guidelines of this section to sketch the curve. y = x2 - 4 y y 5 X 5

Jse the guidelines of this section to sketch the curve. y = x2 - 4 y y 5 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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**Curve Sketching for the Function \( y = \frac{x}{x^2 - 4} \)** **Objective:** Use the given function to sketch the curve using mathematical guidelines. **Function Analysis:** The function to consider is \( y = \frac{x}{x^2 - 4} \). 1. **Vertical Asymptotes:** - Occur where the denominator is zero, which means solving \( x^2 - 4 = 0 \). - This gives \( x = \pm 2 \). Thus, vertical asymptotes are at \( x = 2 \) and \( x = -2 \). 2. **Horizontal Asymptotes:** - Since the degrees of the numerator and denominator are the same, divide the leading coefficients. Here, both are leading terms with coefficients 1, giving a horizontal asymptote at \( y = 0 \). **Graph Descriptions:** - **Top Left Graph:** - Displays the function crossing the x-axis and moving upwards and downwards asymptotically near the lines \( x = 2 \) and \( x = -2 \). - **Top Right Graph:** - Shows a similar pattern but reflects a different function behavior around the asymptotes. - **Bottom Left Graph:** - Demonstrates the function moving in the opposite direction from the top left graph but still respects the same asymptotic boundaries. - **Bottom Right Graph:** - Displays an inverted graph with movements that again differ around the asymptotic lines. Each graph depicts variations of how the curve might interact with the asymptotes, critical value intersections, and boundaries set mathematically by the function equation.
Transcribed Image Text:**Curve Sketching for the Function \( y = \frac{x}{x^2 - 4} \)** **Objective:** Use the given function to sketch the curve using mathematical guidelines. **Function Analysis:** The function to consider is \( y = \frac{x}{x^2 - 4} \). 1. **Vertical Asymptotes:** - Occur where the denominator is zero, which means solving \( x^2 - 4 = 0 \). - This gives \( x = \pm 2 \). Thus, vertical asymptotes are at \( x = 2 \) and \( x = -2 \). 2. **Horizontal Asymptotes:** - Since the degrees of the numerator and denominator are the same, divide the leading coefficients. Here, both are leading terms with coefficients 1, giving a horizontal asymptote at \( y = 0 \). **Graph Descriptions:** - **Top Left Graph:** - Displays the function crossing the x-axis and moving upwards and downwards asymptotically near the lines \( x = 2 \) and \( x = -2 \). - **Top Right Graph:** - Shows a similar pattern but reflects a different function behavior around the asymptotes. - **Bottom Left Graph:** - Demonstrates the function moving in the opposite direction from the top left graph but still respects the same asymptotic boundaries. - **Bottom Right Graph:** - Displays an inverted graph with movements that again differ around the asymptotic lines. Each graph depicts variations of how the curve might interact with the asymptotes, critical value intersections, and boundaries set mathematically by the function equation.
**Guidelines for Sketching a Curve** The following checklist is intended as a guide to sketching a curve \( y = f(x) \) by hand. Each item is relevant to every function, regardless of its specific properties. **A. Domain** Start by determining the domain \( D \) of \( f \), which is the set of values of \( x \) for which \( f(x) \) is defined. **B. Intercepts** - The y-intercept is \( f(0) \) and this tells where the curve intersects the y-axis. - To find the x-intercepts, set \( y = 0 \) and solve for \( x \). **C. Symmetry** 1. **Even Functions**: If \( f(-x) = f(x) \), the curve is symmetric about the y-axis. Examples include \( y = x^2 \) and \( y = \cos x \). - Figure 1(a) illustrates an even function with reflectional symmetry. 2. **Odd Functions**: If \( f(-x) = -f(x) \), the curve is symmetric about the origin. Examples are \( y = x^3 \) and \( y = \sin x \). - Figure 1(b) shows an odd function with rotational symmetry. 3. **Periodic Functions**: If \( f(x + p) = f(x) \) for all \( x \) in \( D \), \( f \) is periodic with period \( p \). \( y = \sin x \) is an example. - Figure 2 displays such a function with translational symmetry. **D. Asymptotes** 1. **Horizontal Asymptotes**: If \(\lim_{x \to \pm \infty} f(x) = L\), the line \( y = L \) is a horizontal asymptote. 2. **Vertical Asymptotes**: If \(\lim_{x \to a^+} f(x) = \pm \infty\) or \(\lim_{x \to a^-} f(x) = \pm \infty\), then \( x = a \) is a vertical asymptote. 3. **Slant Asymptotes**: These are discussed elsewhere. **E. Intervals of Increase or Decrease** Use the first derivative \(
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. Each item is relevant to every function, regardless of its specific properties. **A. Domain** Start by determining the domain \( D \) of \( f \), which is the set of values of \( x \) for which \( f(x) \) is defined. **B. Intercepts** - The y-intercept is \( f(0) \) and this tells where the curve intersects the y-axis. - To find the x-intercepts, set \( y = 0 \) and solve for \( x \). **C. Symmetry** 1. **Even Functions**: If \( f(-x) = f(x) \), the curve is symmetric about the y-axis. Examples include \( y = x^2 \) and \( y = \cos x \). - Figure 1(a) illustrates an even function with reflectional symmetry. 2. **Odd Functions**: If \( f(-x) = -f(x) \), the curve is symmetric about the origin. Examples are \( y = x^3 \) and \( y = \sin x \). - Figure 1(b) shows an odd function with rotational symmetry. 3. **Periodic Functions**: If \( f(x + p) = f(x) \) for all \( x \) in \( D \), \( f \) is periodic with period \( p \). \( y = \sin x \) is an example. - Figure 2 displays such a function with translational symmetry. **D. Asymptotes** 1. **Horizontal Asymptotes**: If \(\lim_{x \to \pm \infty} f(x) = L\), the line \( y = L \) is a horizontal asymptote. 2. **Vertical Asymptotes**: If \(\lim_{x \to a^+} f(x) = \pm \infty\) or \(\lim_{x \to a^-} f(x) = \pm \infty\), then \( x = a \) is a vertical asymptote. 3. **Slant Asymptotes**: These are discussed elsewhere. **E. Intervals of Increase or Decrease** Use the first derivative \(
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