Math Problem Statement
Let R(x) =( x2 + x − 1)/(3x^2 +11x -4). In order to sketch the graph of y = R(x),
Solution
To sketch the graph of the rational function , we need to analyze several aspects of the function:
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Domain: Determine where the function is defined by finding the values of that make the denominator zero: Using the quadratic formula : This gives us the roots: Thus, the function is undefined at and .
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Intercepts:
- Y-intercept: Set :
- X-intercepts: Set the numerator equal to zero: Using the quadratic formula:
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Asymptotes:
- Vertical asymptotes occur at the values where the denominator is zero: and .
- Horizontal asymptote: Since the degrees of the numerator and denominator are both 2, the horizontal asymptote is given by the ratio of the leading coefficients:
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Behavior Near Asymptotes:
- Analyze the behavior of as approaches the vertical asymptotes from the left and right.
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End Behavior: As , approaches .
Summary of Graph Features:
- Vertical Asymptotes: and
- Horizontal Asymptote:
- Y-intercept:
- X-intercepts: and
Graphing Steps:
- Plot the intercepts and asymptotes.
- Sketch the behavior of the function in each interval determined by the asymptotes.
- Ensure the function approaches the horizontal asymptote as goes to infinity.
Would you like to delve into any specific aspect of this function, or do you have any questions? Here are some follow-up questions:
- How do you find the derivative of ?
- Can you explain the significance of vertical and horizontal asymptotes?
- What happens to as approaches the asymptotes?
- How would you graph this function using technology?
- Can you provide a numerical example of evaluating at specific points?
Tip: Always check for intercepts and asymptotes when sketching rational functions, as they provide key information about the graph's behavior.
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Math Problem Analysis
Mathematical Concepts
Rational Functions
Graphing
Algebra
Formulas
Quadratic formula: x = (-b ± √(b² - 4ac)) / 2a
Horizontal asymptote: y = leading coefficient of numerator / leading coefficient of denominator
Theorems
Asymptote Theorems
Properties of Rational Functions
Suitable Grade Level
Grades 10-12
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