Graph the function y = x3 + 3x2 – x – 3. which lists all of the turning points of the graph? (–3, 0) and (1, 0) (–2, 3) and (0, –3) (–2, 3), (–1, 0), and (0, –3) (–3, 0), (–1, 0), and (1,0)

Answer with explanation:

The given function is:

  y=x³+3 x²-x-3

    =x²(x+3) -1(x+3)

    = (x²-1)(x+3)

   = (x-1)(x+1)(x+3)

Put, y=0 , gives

⇒ (x-1)(x+1)(x+3)=0

≡→x-1=0 ∧ → x+1=0 ∧→x+3=0

≡x=1, -1, -3

So, there are three breaking points in the curve.

1.⇒Putting, x=1 , in the function

y(1)=1³+3×1²-1 -3

  =1 +3-4

  =0

2. ⇒Putting, x=-1 , in the function

y(-1)=(-1)³+3×(-1)²-(-1) -3

  = -1 +3+1-3

  =0

3.⇒Putting, x=-3 , in the function

y(-3)=(-3)³+3×(-3)²-(-3) -3

  =-27+27+3-3

  =0

So, These are points where curve crosses the X axis, which are , (1,0),(-1,0) and (-3,0).

But turning points are those points where curve takes the turn , that is, if it is increasing function then it starts decreasing or becomes constant function after that point, and if it is a decreasing function then it becomes increasing or constant function.

So,turning points of the given function by looking at the graph of the function  by approximation

Option B:→(-2,3) and (0, -3)

expression #3 is not equivalent. the others are.