Answer:
y = (x +1)(x -1)²(x -4)²
Explanation:
You want the factored form of an equation for the function shown in the graph.
Zeros
The graph has a "zero" where it touches or crosses the line y=0 (the x-axis). Those points are x = -1, x = 1, and x = 4.
Each zero of value 'p' corresponds to a factor (x -p). Where the graph touches, but does not cross the x-axis, the zero has "even multiplicity". That means its exponent in the factored form will be an even number greater than 0.
The curve of the function at x=1 and x=4 tells you the multiplicity of those zeros is 2 (not 4 or more).
Factored form
The product of the known factors is ...
y = (x +1)(x -1)²(x -4)²
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Additional comment
The greater the multiplicity of the zero, the more the graph "hugs" the x-axis in its vicinity.
The function generally increases, indicating it is of odd degree. It has 4 turning points, so is at least degree 5. These observations are consistent with the factors we found.
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