ƒ(x) = [−2(x +6 )(x−4)] / [ (x+5)(x−2)] = (48−4x−2x2) / (x2 +3x−10)
Another one could be g(x) = ƒ(x)( x2+3) / (x2+3) = (144−12x+42x2−4x3−2x4 ) / (x4 +3x3−7x2 +9x−30)
g(x) = (144−12x+42x2−4x3−2x4 ) / (x4 +3x3−7x2 +9x−30)
You can make an infinite number of rational functions that satisfy the initial conditions...
Adam B.
tutor
Hi Doug,
I am a great fun of your answers. You are having the gift of teaching. Take it from a veteran..
I have the same question and I am working on it. Talk to you later
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08/18/21
Doug C.
Check it out--not sure if this can be extended in a general way, but for this problem all requirements are met and the y-intercept is at (0,-5). desmos.com/calculator/oi06x0rvos
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08/20/21
Adam B.
tutor
I will
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08/20/21
Adam B.
tutor
What would the value of a be if I wished a y-intercept of (0,-7) ?
And what about for an additional factor of ( x^4 + 69 ) ?
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08/20/21
Doug C.
desmos.com/calculator/ohulyv1bte Graph modified to allow setting y-coordinate of y-intercept with a slider and the value of "a" depends on k. Turns out for this function a= -125k/24 This was determined by letting x = 0, y = k in function definition and solving for a. An additional factor of x^4+b would change the function to no horizontal asymptote.
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08/20/21
Doug C.
I see a problem, k must be negative else new roots are introduced.
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08/21/21
Adam B.
tutor
Doug, I meant the additional factor would appear in both numerator and denominator
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08/21/21
Doug C.
I wonder if there is a way to modify the function definition so that it satisfies all the given requirements (with the exception of f(-2) = -4), but add the additional requirement that the y-intercept is (0,-5), for example) (as opposed to the current y-intercept of (0,-4.8). Or what if the last requirement was modified to f(-2) = -3 -- could such a function be created? I actually had a student ask this question a lot of years ago, and I never did find an answer.08/18/21