Question 233182: Hello.
First of all I want to thank you in advance.
I need to list coordinates of the intercepts at which the graph of the
following function crosses the x-axis.
Function:
By just looking at the function I
came up with (-1,0),(2,0),(3,0)
My professor listed (-1,0),(2,0)
as correct answers.
Isn't (3,0) part of the coordinates?
Thank you!
William
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! As far as I can tell, the crossing points are:
x = -1, x = 2, x = 3
This would be calculated as follows:
Set the equation to 0 to get:
2(x+1)(x-2)^3(x-3)^2 = 0
Divide both sides by 2 to get:
(x+1)(x-2)^3(x-3)^2 = 0
This equation will be true if:
x+1 = 0 or (x-2)^3 = 0 or (x-3)^2 = 0
This makes:
x = -1 or x = 2 or x = 3
If you plug any one of these values into the equation, then the equation will be true which means that the graph of the equations touches the x-axis at those points.
You said:
By just looking at the function I
came up with (-1,0),(2,0),(3,0)
My professor listed (-1,0),(2,0)
as correct answers.
Isn't (3,0) part of the coordinates?
Looks like you are correct in that the equation touches the x-axis at those points.
It appears your professor might also be correct IF he is stating that the equation does not cross the x-axis at those points.
It touches at x = 3, but it does not cross.
Check the following graphs out to see what I mean:
The graph on the left is a long range view.
The graph on the right is a close in view.
Graph of your equation
It may be semantics, but it does not appear that the equation CROSSES the x-axis at x = 3, even though it TOUCHES the x-axis at x = 3.
I believe that's what he's saying.
If you were solving for the roots of the equation, which is defined as the point where the equation crosses the axis, then you would be correct in including x = 3. Otherwise a quadratic equation that has only one root would not be valid because it technically doesn't cross the x-axis at that point, but touches it, as is the case with this equation.
That's the only reason for the discrepancy that I can think of.
I would say that x = 3 IS an intercept, but that's only my opinion. The graph definitely TOUCHES the x-axis at x = 3 and that's a valid intercept point when dealing wth quadratic equations from everything I've seen so far.
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