Math Problem Statement
Solution
To create a rational function with the specified properties, let’s analyze each requirement:
-
Vertical asymptotes at and :
Vertical asymptotes are created by the zeros in the denominator. Thus, the denominator should have factors of and . -
X-intercepts at and :
X-intercepts occur when the numerator is zero. Therefore, the numerator should have factors of and . -
Y-intercept at 4:
To find the y-intercept, we substitute into the function. This will allow us to determine any constant factor required in the function.
General Form of the Function
Given the above factors, we can start with the following form:
where is a constant that we will determine using the y-intercept condition.
Finding
To satisfy the y-intercept at , substitute and set :
Simplify this expression:
Multiply both sides by :
Final Equation
Now that we have , the equation for the rational function is:
This function has the specified vertical asymptotes, x-intercepts, and y-intercept.
Would you like more details or have any questions?
Here are 5 related questions for further exploration:
- How would the equation change if the vertical asymptotes were at different values?
- What happens to the graph if we change the x-intercepts or the y-intercept?
- How would adding a horizontal asymptote affect the equation?
- What impact does the value of have on the function's graph?
- Can we create a similar function with horizontal asymptotes as well?
Tip: When finding constants in rational functions, use given points (like y-intercepts) to solve for any unknowns.
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Math Problem Analysis
Mathematical Concepts
Rational Functions
Asymptotes
Intercepts
Formulas
General form of a rational function: y = k * (numerator factors) / (denominator factors)
Theorems
Intercept Theorem for rational functions
Asymptote Properties in Rational Functions
Suitable Grade Level
High School (Grades 10-12)
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