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Given the function f(x)=x(x-2)^2(x-4)
I. find x and y intercept
II.Determine whether the graph crosses or touches the x-axis at each intercept
III. Graph the function
set y=0, then solve for x
x-intercepts=0,2(mult. 2), and 4
set x=0, then solve for y
II. Number line:
For x>4, f>0
Moving to the left, the sign of each interval switches each time we go thru a zero of odd multiplicity like 1, 3, 5, etc. The sign of each in interval does not change if we go thru a zero of even multiplicity like 2, 4, 6, etc. Accordingly, the graph crosses the x-axis at x= 0 and 4, but at x=2, only touches but does not cross the x-axis.
III.See graph below as a visual check on the above: