2022-03-09
RizerMix
Expert2023-04-23Added 656 answers
To create a rational function with four asymptotes and three x intercepts, we can start by using the formula:
where a is a constant, are the x intercepts, and are the asymptotes. To achieve the desired result, we need to choose appropriate values for these parameters.
Let's start by choosing the asymptotes. As the function has four asymptotes, we will use the equation of a hyperbola with vertical and horizontal asymptotes:
Next, we need to select the x-intercepts. To make sure that the function has exactly three x-intercepts, we will choose three distinct real numbers:
Finally, we need to determine the value of the constant a. We can do this by setting the leading coefficient of the function to 1. The leading coefficient is the coefficient of the highest-degree term in the numerator, which is . Since this term has degree 3, we need to choose a such that:
where b is a constant that we can choose arbitrarily. Let's set b = 1 for simplicity.
Substituting all the values we have chosen, we get the final rational function:
This function has four vertical and horizontal asymptotes at x = 2, x = -2, x = 3i, and x = -3i, and three x-intercepts at x = -1, x = 0, and x = 1.
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