SOLUTION: What is the x- and y- axis of f(x)=(x-4)(x+2)^2(x-2) Also, does the graoh of f touch or cross the x-axis at the x-intercept?

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Question 821326: What is the x- and y- axis of f(x)=(x-4)(x+2)^2(x-2)
Also, does the graoh of f touch or cross the x-axis at the x-intercept?
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Read the zeros directly from the function definition and use them as critical points which define intervals along the x-axis.
The zeros are at -2, 2, and 4. You want to check a value for sign in each of the intervals:
x%3C-2;
-2%3Cx%3C2;
2%3Cx%3C4;
4%3Cx.

Intuitively, you would well guess that f(x) will be positive in x%3C-2 and in 4%3Cx.

Try 0 within -2%3Cx%3C2:
%280-4%29%280%2B2%29%5E2%280-2%29=%28-4%29%284%29%28-2%29%3E0, so positive.

Try 3 within 4%3Cx:
%283-4%29%283%2B2%29%5E2%283-2%29=%28-1%29%2825%29%281%29%3C0, negative.

You should sketch a graph based on these and you then can find where f touches or crosses the x-axis. Note what happens around x=-2; positive on both sides near this, so f not cross the x-axis, but TOUCHES it only.

graph%28300%2C300%2C-5%2C5%2C-25%2C30%2C%28x-4%29%28x%2B2%29%5E2%28x-2%29%29