determine the graphs end behavior find the x and y intercept

determine the graphs end behavior, find the x and y intercepts, determine whether the graph has symmetry. determine the graph of the function
f(x) = x^3 - 3x^2 - x + 3

find the y intercept by setting x equal to zero and computing f(0)

determine whether the graph has y axis symmetry, origin symmetry, or neither

determine the graph of the function. use the fact that the maximum number of turning points of the graph is n- 1, with n as the leading term to check the correct choice

Solution

x^3 = leading term
So, we have odd degree and positibve lead term
So, end behavior is : falls on left, rises on right

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y-int :
Plug in x= 0, we get y = 3

So, y-int is the point (0,3)

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x-int :
x^3 - 3x^2 - x + 3 = 0

Lets factor by grouping and pairing them together :
x^2(x - 3) - 1(x - 3) = 0

(x - 3)(x^2 - 1) = 0

(x - 3)(x - 1)(x + 1) = 0

x = -1 , 1 and 3

So, xint are (-1,0) , (1,0) and (3,0)

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Symmetry :
No symmetry along y-axis or origin because this function is neither odd nor even

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n = degree = 3

So, max number of turning points = n - 1 = 3 - 1 = 2 turning points