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Write an equation for the function graphed below. (Hint: Use the y-intercept to find the leading coefficient.) y=

Write an equation for the function graphed below. (Hint: Use the y-intercept to find the leading coefficient.) y=

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Write an equation for the function graphed below. (Hint: Use the y-intercept to find the leading coefficient.)

y=

 

The image displays a graph with two vertical asymptotes and three regions of graph behavior. The vertical asymptotes are indicated by red dashed lines at \( x = -1 \) and \( x = 4 \). ### Description of the Graph: 1. **Left Region (\( x < -1 \)):** - The curve approaches the vertical asymptote at \( x = -1 \) from the left. - As \( x \) decreases, the function rises sharply and approaches negative infinity. 2. **Middle Region (\( -1 < x < 4 \)):** - The curve starts from negative infinity just above \( x = -1 \), increasing rapidly to positive values. - It crosses the x-axis between \( x=0 \) and \( x=1 \). - As it approaches the vertical asymptote at \( x = 4 \), the curve decreases again and moves towards negative infinity. 3. **Right Region (\( x > 4 \)):** - The curve approaches the vertical asymptote at \( x = 4 \) from the right, starting from positive infinity. - As \( x \) increases, the curve falls gradually, approaching the x-axis but remaining above it. ### Axis Details: - **X-Axis:** Marks are plotted at intervals of 1 from \( x = -7 \) to \( x = 7 \). - **Y-Axis:** Marks are plotted at intervals of 1 from \( y = -6 \) to \( y = 5 \). The graph represents a rational function experiencing vertical asymptotic behavior at \( x = -1 \) and \( x = 4 \), indicating undefined points at these x-values.
Transcribed Image Text:The image displays a graph with two vertical asymptotes and three regions of graph behavior. The vertical asymptotes are indicated by red dashed lines at \( x = -1 \) and \( x = 4 \). ### Description of the Graph: 1. **Left Region (\( x < -1 \)):** - The curve approaches the vertical asymptote at \( x = -1 \) from the left. - As \( x \) decreases, the function rises sharply and approaches negative infinity. 2. **Middle Region (\( -1 < x < 4 \)):** - The curve starts from negative infinity just above \( x = -1 \), increasing rapidly to positive values. - It crosses the x-axis between \( x=0 \) and \( x=1 \). - As it approaches the vertical asymptote at \( x = 4 \), the curve decreases again and moves towards negative infinity. 3. **Right Region (\( x > 4 \)):** - The curve approaches the vertical asymptote at \( x = 4 \) from the right, starting from positive infinity. - As \( x \) increases, the curve falls gradually, approaching the x-axis but remaining above it. ### Axis Details: - **X-Axis:** Marks are plotted at intervals of 1 from \( x = -7 \) to \( x = 7 \). - **Y-Axis:** Marks are plotted at intervals of 1 from \( y = -6 \) to \( y = 5 \). The graph represents a rational function experiencing vertical asymptotic behavior at \( x = -1 \) and \( x = 4 \), indicating undefined points at these x-values.
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