Find a possible formula for the polynomial function f graphe

Find a possible formula for the polynomial function f graphed below. Identify f as odd or even degree based upon its long-run behavior. Identify the zeros of f and the multiplicity of zeros (if any). Let a_n represent a stretch factor. Use the fact that f(0) = -3. Find a_n Determine a possible formula for this polynomial.

Solution

a)
Notice that on both the left and right,
the graph is veering off in opposite
directions
So, this is an ODD DEGREE function

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b)
The zeros are where the curve meets the x-axis
and this happens at x = =-1 and 3
Notice that at x = -1, the curve passes thru smoothly
And at x = 3, the curve rebounds, so multiplicity is even

So, zeros :
x = -1 (multiplicity of one)
x = 3(multiplicity of two)

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c) Stretch factor is :
Compressed by a factor of 1/3

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d)
We have x = -1 multiplicity of one
So, this is the term (x+1)

And we have x =3, this will be (x-3)
But it has multiplicity of two
So, (x-3)^2

So, the formula is :
(x+1)(x-3)^2

But as x ---> -inf , y ---> +inf
And as x---> +inf , y ---> -inf

So, this will have a negative leading term

So, the formula is :
-(x + 1)(x - 3)^2

And we also have f(0) = -3
So, we gotta multiply the function above by 1/3

So, final formula is :

-1/3 * (x + 1) * (x - 3)^2 ------> ANSWER