Math Problem Statement
Solution
To solve for the equation of , let’s analyze the information given:
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Vertical Asymptotes: The function has vertical asymptotes at and . This suggests that the denominator of should include factors and .
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Horizontal Asymptote: The function has a horizontal asymptote at . For a rational function to have a horizontal asymptote at zero, the degree of the numerator must be less than the degree of the denominator. This implies that the numerator should be a constant or a first-degree polynomial.
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X-Intercept: The function has an -intercept at . For a rational function to have an -intercept at , the numerator should include as a factor.
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Point (2,1): The function passes through the point , which we can use to solve for any constant in the equation.
Choosing the Form of
Given the information, a suitable form for would be: since this form accommodates two vertical asymptotes and an -intercept.
Finding the Value of
To determine , we use the point : Simplifying:
Final Equation
The equation for is:
Let me know if you’d like further explanation or have questions.
Here are five related questions to consider:
- How would the equation change if there were an additional vertical asymptote at another -value?
- What would the function look like if it had a different horizontal asymptote, such as ?
- How does the degree of the polynomial in the numerator affect the horizontal asymptote?
- Why does the factor in the numerator result in an -intercept at ?
- How can you verify that the function passes through a specific point?
Tip: When determining a rational function, identify vertical asymptotes from the denominator and -intercepts from the numerator factors.
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Math Problem Analysis
Mathematical Concepts
Rational Functions
Asymptotes
Intercepts
Formulas
f(x) = a(x - 1) / ((x + 1)(x - 3))
Theorems
Properties of Rational Functions
Suitable Grade Level
Grades 11-12
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