Use the function f(x) = −16x2 + 24x + 16 to answer the questions. Part A: Completely factor f(x). (2 points) Part B: What are the x-intercepts of the graph of f(x)? Show your work. (2 points) Part C: Describe the end behavior of the graph of f(x). Explain. (2 points) Part D: What are the steps you would use to graph f(x)? Justify that you can use the answers obtained in Part B and Part C to draw the graph. (4 points)

see below

Step-by-step explanation:

f(x) = −16x^2 + 24x + 16

Factor out -8

f(x) = -8( 2x^2-3x-2)

Factor

f(x) = -8 (x-2) ( 2 x+1)

Set equal to zero to find the x intercepts

0 =  -8 (x-2) ( 2 x+1)

Using the zero product property

x-2 =0   2x+1 =0

x=2     2x=-1

x=2   x = -1/2

The x intercepts are

(2,0)  and (-1/2,0)

The end behavior

Let x = -∞

f(x) = −16( -∞)^2 + 24( -∞) + 16

      The squared term dominates

     =  −16( -∞)^2 =   −16( ∞) =   −( ∞)

It goes to  -∞ as x goes to ( -∞)    

Let x = ∞

f(x) = −16( ∞)^2 + 24( ∞) + 16

      The squared term dominates

     =  −16( ∞)^2 =    −16( ∞) =   −( ∞)

It goes to  -∞ as x goes to ( ∞)    

We need the vertex, the zeros and the end behavior to graph the parabola

The vertex is  1/2 way between the zeros

(-1/2+2) /2 = 3/4

f(3/4) =  -8 (3/4-2) ( 2 *3/4+1) = 25

We know it opens down since the x^2 term has a negative coefficient