A quadratic function is given. f(x) = −x2 + 4x + 2 (a) Express the quadratic fun

Algebra
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A quadratic function is given.

f(x) = −x2 + 4x + 2
(a) Express the quadratic function in standard form. 
f(x) =

(b) Find its vertex and its x- and y-intercept(s). (If an answer does not exist, enter DNE.) 
vertex    (xy) =
 
 
x-intercepts    (xy) =
 
 
 (smaller x-value)
(xy) =
 
 
 (larger x-value)
y-intercept    (xy) =
 
 
Jul 6th, 2015

Hello!

(a) f(x) = -x^2 + 4x + 2 = -(x^2 - 4x) + 2 = -[(x-2)^2 -4] + 2 = -(x-2)^2 + 6. It is the standard form (perfect square is picked out).

(b)

the vertex is where the perfect square is zero, i.e. at x=2. For this x y=6. (x, y) = (2, 6).

the x-intercepts are roots, f(x) = -(x-2)^2 + 6 = 0. So (x-2)^2 = 6, x-2 = +-sqrt(6), x = 2 +- sqrt(6). y of course 0:
(x, y) = (2-sqrt(6), 0)   (smaller x-value)
(x, y) = (2+sqrt(6), 0)    (larger x-value)

(sqrt is for square root)

the y-intercept is at x=0, f(0) = 2.
(x, y) = (0, 2).

Please ask if anything is unclear.
Jul 6th, 2015

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