Math Problem Statement
Write an equation for a rational function with: Vertical asymptotes at x = − 5 and x = 1 x -intercepts at ( − 4 , 0 ) and ( − 1 , 0 ) y -intercept at ( 0 , 7 )
Solution
To construct a rational function with the given properties, let’s break it down step by step.
1. General form of a rational function:
A rational function has the general form:
where is the numerator (containing the x-intercepts) and is the denominator (containing the vertical asymptotes). Let's build these based on the conditions.
2. Vertical asymptotes:
- Vertical asymptotes occur when the denominator becomes zero.
- Since the vertical asymptotes are at and , the factors of the denominator must be:
3. x-intercepts:
- The numerator determines the x-intercepts.
- Since the x-intercepts are at and , the numerator must contain these as factors:
4. Putting it together:
So far, the rational function is:
where is a constant that ensures the function passes through the y-intercept at .
5. Finding the constant :
- Use the y-intercept at to find .
- Substituting and :
6. Final equation:
Thus, the rational function is:
Or equivalently:
Would you like further explanation or a graph of this function?
Here are 5 related questions:
- What happens to the function as ?
- How do horizontal asymptotes differ from vertical asymptotes?
- How do you find slant asymptotes for rational functions?
- How would the equation change if the x-intercepts had multiplicity 2?
- How can you determine whether the function has any holes?
Tip: When solving for constants using known points, be mindful of negative signs—small mistakes can affect the solution significantly.
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Math Problem Analysis
Mathematical Concepts
Rational Functions
Asymptotes
Intercepts
Formulas
Rational function form f(x) = N(x) / D(x)
Vertical asymptotes: set denominator factors to zero
x-intercepts: set numerator factors to zero
Substitute known point to solve for constant
Theorems
Concept of Vertical Asymptotes
Concept of Intercepts
Suitable Grade Level
Grades 11-12
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