Problema Solution
For the function f(x) = -2(x - 6)(x - 3)
What are the x-intercepts? What is the y-intercept? What is the Axis of Symmetry? What is the Vertex? Describe the max or min value of the function?
Answer provided by our tutors
consider that f(x)=-2(x - 6)(x - 3) = -2(x^2-6x-3x+18)= -2x^2+12x+6x-36 =-2x^2+18x-36
define: a=-2, b=18, c=-36
X-INTERCEPTS where:
x1 = (-b+sqrt(b^2-4ac))/2a = (-18+sqrt(18^2-4*(-2)*(-36)))/(-4)=(-18+sqrt(324-288))/(-4)=(-18+6)/(-4)=-12/-4 = x1=3
x2 = (-b-sqrt(b^2-4ac))/2a = (-18-sqrt(18^2-4*(-2)*(-36)))/(-4)=(-18-sqrt(324-288))/(-4)=(-18-6)/(-4)=-24/-4 = x2=6
Y-INTENCEPTS where y=c --> y=-36 -->(0,-36)
function is symmetric by x-axe where x=-b/2a = -18/2(-2) = -18/ -4 = 9/2 = 4,5
VERTEX has coords x= -b/2a = 9/2 like above and y= f(9/2) = -2(9/2 - 6)(9/2 - 3) = -2(-3/2)(3/2)=-2(-9/4)=18/4=9/2 so VERTEX has coords (9/2,9/2)
Consider the derivate of f(x)=f'(x)=-4x+18, we have specials points where f'(x)=0 --> where -4x+18=0 -> -4x=-18 -> x=9/2.
The concavity of the function is under side so (9/2,9/2) is a MAX-VALUE