Question 969595: Sketch a graph and find one possible equation for a polynomial function f(x) with x intercepts at -2 and 1, a y intercept at 5 and where f(x) = -infinity and f(x)=infinity.
I came up with f(x)= -(5/4)x^cubed - (15/4)x^2 +5. Not sure if this is correct, but it does make a graph that looks correct. Is this a possible equation for this scenerio?
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! Yes, it is, IF the question says that the range is between minus infinity and plus infinity. The solutions are exact integers.
You would need a cubic equation for this, because a quadratic could have the intercepts right, but f(x) as x became very large or very small would be positive or negative infinity.
You have another way to do this, so let me explain it better than I did.
It's a cubic, and two intercepts have to be the same, since there are 3 roots and only 2 distinct intercepts.
There can be two roots at 1 and one root at minus 2. Root at -2 is (x+2), for when (x+2)=0, x=-2.
The root at +1 is (x-1). We will square that one (use it twice).
f(x)=a(x-1)^2 (x+2)
What is a? Well, use the y-intercept OR any point they give you.
(0,5) is the point you were given. Put that into the function.
5=a (-1^2)*2 When x becomes 0 on the right, it is (-1^2)*2) or 2.
5=2a
a=5/2
You now have f(x)= (5/2)(x-1)^2(x+2)
f(x)=(5/2) * (x^2-2x+1) (x+2) If I multiply this out
f(x)=(5/2)(x^3-3x+2)
Here is the graph below.
I hope that helps. I should have done it this way to show you that both functions worked. You got the 5/4, because you squared the -2 and multiplied it by 1. I got 5/2 because I squared the -1 and multiplied it by 2.
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