Math Problem Statement
Solution
The graph shown suggests we need to find the rational function by analyzing the following key features:
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Vertical asymptotes: These occur where the denominator of the rational function equals zero. Based on the graph, we can see vertical asymptotes at and . This suggests factors in the denominator of the form and .
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Horizontal asymptote: There seems to be a horizontal asymptote at , which suggests the degree of the numerator is less than or equal to the degree of the denominator.
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Behavior near vertical asymptotes:
- At , the graph approaches positive infinity on the left and negative infinity on the right.
- At , the graph approaches negative infinity on the left and positive infinity on the right.
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Zeros of the function: The graph crosses the x-axis at , suggesting a zero at . This means the numerator contains a factor .
Possible Rational Function
From this, we can hypothesize that the function is of the form:
Determining the constant :
- We also observe a point passing through on the graph. Substituting and into the function will allow us to solve for .
Let's calculate the value of .
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Math Problem Analysis
Mathematical Concepts
Rational Functions
Asymptotes
Zeros of Functions
Formulas
Rational function: f(x) = a(x - 2) / ((x + 2)(x - 4))
Theorems
Vertical Asymptote Theorem
Horizontal Asymptote Theorem
Zero of a Function Theorem
Suitable Grade Level
Grades 10-12
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