40 points for this question! algebra 2. with all explanation, !

You will need three roots for this, so we have
 Let x = -30, -10 and +20
So the factors will be  (x+30)(x+10)(x-20)
The divide it to 100, this will help bring the peak up and down
So the polynomial function R(x) will become
1/100 *  (x+30)(x+10)(x-20)
R(x) = 1/100 * (x+30)(x+10)(x-20)
 
Finding the X-intercept:
Let R(x) = 0 and solve for x.
1/100 * (x+30)(x+10)(x-20) = 0
x = -30, -10, 20 are the x-intercepts.

vertex form is f(x) = ( x-1)² + 3.

Step-by-step explanation:

Given :  f(x) = 4 + x² – 2x.

To find : Which function in vertex form is equivalent to f(x) = 4 + x² – 2x.

Solution : We have given that  f(x) = 4 + x² – 2x.

Rearrange in function :  f(x) = x² – 2x + 4.

Vertex form of y = a(x-h)²+k .

Now, add and subtract (-1)² in f(x) .

f(x) = x² – 2x + 4 + (-1)²- (-1)².

By  Squaring formula : (x +a)²= x²+ a²+2ax .

We can write x² – 2x + (-1)²- (-1)² +4 as ( x-1)²- (-1)² +4.

f(x) = ( x-1)²- (-1)² +4.

Now , completing square ,

f(x) = ( x-1)²- 1 +4.

f(x) = ( x-1)² + 3.

Therefore , vertex form is f(x) = ( x-1)² + 3.