Math Problem Statement
Write a rational function that has the following characteristics: The only zero is 4; asymptotes at x=-2, x=3, and y= (5/3)
Solution
We are asked to write a rational function that satisfies the following conditions:
- The only zero is at .
- Vertical asymptotes at and .
- A horizontal asymptote at .
Step-by-Step Construction:
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Zero at : The function must have a factor of in the numerator, since the zero occurs when .
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Vertical asymptotes at and : The denominator must have factors of and because vertical asymptotes occur where the denominator is zero but the numerator is not.
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Horizontal asymptote at : To achieve a horizontal asymptote of , the degrees of the numerator and denominator should be the same, and the ratio of the leading coefficients of the numerator to the denominator should be .
Rational Function:
A function satisfying these conditions is:
This function meets all the given conditions:
- It has a zero at because the numerator becomes zero when .
- It has vertical asymptotes at and because the denominator becomes zero at these points.
- The horizontal asymptote is , because for large values of , the function behaves like , which simplifies to as .
Would you like more details on the graph or the function's behavior?
Here are some related questions you might explore:
- How does the degree of the numerator affect the horizontal asymptote?
- What happens if we change the location of the vertical asymptotes?
- How would the function change if the horizontal asymptote were instead?
- How does the multiplicity of zeros and poles (asymptotes) affect the shape of the graph?
- What if there were no horizontal asymptote, but instead a slant asymptote?
Tip: When graphing rational functions, always start by identifying the asymptotes and zeros for a clearer understanding of the overall behavior.
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Math Problem Analysis
Mathematical Concepts
Rational Functions
Zeros of Functions
Asymptotes
Formulas
Rational Function: f(x) = (numerator)/(denominator)
Horizontal Asymptote: Ratio of leading coefficients
Vertical Asymptote: Denominator = 0 when the numerator ≠ 0
Theorems
Horizontal Asymptote Theorem
Vertical Asymptote Theorem
Zero of a Function Theorem
Suitable Grade Level
Grades 10-12
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