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Three Intercepts (Posted on 2006-06-19) Difficulty: 3 of 5
Given the information that the graph of a function has a y-intercept at (0,1) and exactly two x-intercepts at (2,0) and (4,0), how many different functions can you find that pass through these three points?

Note: there are infinite families of functions such as high degree polynomials which pass through them, so a single example would suffice for them. Also disallowed would be piecewise function and functions with artificially restricted domains.

No Solution Yet Submitted by Jer    
Rating: 2.0000 (2 votes)

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Solution Polynomial Solutions Comment 7 of 7 |
There is an infinite number of polynomials which satisfy the conditions for the intercepts.

The quadratic polynomial f(x) = (1/8)x^2 - (3/4)x + 1 = 0 is the unique quadratic which passes through the three points given.  Let g(x) be any even degree polynomial which is strictly positive and passes through (0,1).  Then f(x)*g(x) is also an answer to the problem.

For example, g(x) = (1/32)x^2 + (1/4)x + 1.  Then f(x)*g(x) = (1/256)x^4 + (1/128)x^3 - (1/32)x^2 - (1/2)x + 1 is an answer.

  Posted by Brian Smith on 2016-07-09 14:54:42
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