Math Problem Statement
how do I solve this?
Solution
Let's analyze the problem step-by-step:
(i) Identify the horizontal and vertical asymptotes (if any). Explain how you would find them mathematically.
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Vertical Asymptotes:
Vertical asymptotes occur when the denominator of the rational function equals zero (and the numerator does not cancel it out). From the graph:- The vertical asymptotes appear at and , as the graph approaches infinity near these -values.
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Horizontal Asymptotes:
Horizontal asymptotes depend on the degrees of the numerator () and denominator () of the rational function:- If , the horizontal asymptote is .
- If , the horizontal asymptote is .
- If , there is no horizontal asymptote.
From the graph, as , the curve approaches , indicating a horizontal asymptote at .
(ii) Identify the zeros of the rational function.
Zeros occur when the numerator equals zero (and the denominator is non-zero). From the graph:
- The zeros of the function are and , as these are the points where the graph crosses the -axis ().
(iii) Identify the rational function.
To construct the rational function:
- Zeros: The numerator must have factors corresponding to the zeros at and . Hence, the numerator is .
- Vertical Asymptotes: The denominator must have factors corresponding to the vertical asymptotes at and . Hence, the denominator is .
- Horizontal Asymptote: The horizontal asymptote is . This implies the degrees of the numerator and denominator are the same, and the leading coefficients of both are equal. Therefore, no additional constants are needed.
The rational function is:
Would you like me to expand on the derivations or clarify any part? Here are related questions to deepen understanding:
- How do you confirm that is the horizontal asymptote mathematically?
- Why do vertical asymptotes occur at and ?
- What would happen if a zero and an asymptote occur at the same -value?
- How can you verify the zeros and satisfy the equation ?
- How would the graph change if there were an additional vertical asymptote?
Tip: Always verify the symmetry and behavior near asymptotes when sketching or analyzing rational functions.
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Math Problem Analysis
Mathematical Concepts
Rational Functions
Asymptotes (Vertical and Horizontal)
Zeros of Functions
Formulas
Horizontal asymptote rules: y = 0 if n < m, y = leading coefficient ratio if n = m, no horizontal asymptote if n > m
Vertical asymptotes occur when the denominator equals zero
Theorems
Properties of Rational Functions
Suitable Grade Level
Grades 10-12