Question

Use the function f(x) = −16x^2 + 22x + 3 to answer the questions.

Answer 1

Answer:

see below

Step-by-step explanation:

f(x) = −16x^2 + 22x + 3

Factor out the negative

f(x) = -( 16x^2 -22x -3)

      = -(8x+1)(2x-3)

Find the x intercepts

Set y = 0

0 = -(8x+1)(2x-3)

Using the zero product property

8x+1 =0     2x-3 = 0

8x = -1           2x = 3

x = -1/8             x =3/2

The x intercepts are ( -1/8, 0)  and ( 3/2, 0)

The end behavior

-16 x^2 is the dominate term

Let x →-∞

f(-∞) = -16 (-∞)^2 = -16 (∞) = -∞

As x goes to negative infinity  y goes to - infinity

Let x →∞

f(∞) = -16 (∞)^2 = -16 (∞) = -∞

As x goes to  infinity  y goes to - infinity

We know this is a downward facing parabola  a < 0 and this is a quadratic

We have the x intercepts

We can find the axis of symmetry from the zeros

(-1/8+ 3/2) /2 = (-1/8 + 12/8)/2 = (11/8)/2 = 11/6

The axis of symmetry is x = 11/16

Using the axis of symmetry and the equation, we can find the maximum point

y =  -(8*11/16+1)(2*11/16-3) = 169/16

The vertex is at (11/16, 169/16(