Answer:
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
