How do you draw the parabola #y=3x(x+2)#?
↳Redirected from
"What is the equation of a neutralization reaction between sodium hydroxide and hydrochloric acid?"
graph{y=3x(x+2) [-9.5, 10.5, -3, 7]}
#y=3x(x+2)#
Distribute the #3x# into the parenthesis
#y=3x^2+6x#
For the quadratic #y = ax^2 + bx + c# the vertex #(h, k)# which is the same as #(x,y)# is found by solving for #h = –b/(2a)#
#y=3x^2+6x#
#a=3#
#b=6#
#c=0#
#h=-6/(2(3))=-6/6=-1#
#h=-1#
#(h,k) => (-1,k)#
Now, to find the value of #k# (your #y# value), plug the found #h# value #(-1) # into the original equation # y=3x^2+6x#
#y=3(-1)^2+6(-1)#
#y=3(1)-6#
#y=3-6#
#y=-3#
#(h,k)=>(-1,-3)#
#(-1,-3)# is the vertex of the equation
Expansion on prior answers!
I wanted to give a quick derivation of why the vertex point is at #x = -b/(2a) # for anyone who understands differential calculus:
The vertext point is also know as the stationary point, where the functions gradient is #0#
Let #y = ax^2 + bx + c #
Using power rule:
# (dy)/(dx) = 2ax + b #
When gradient is 0:
#2ax + b = 0 #
#=> 2ax = -b #
#=> x = -b/(2a) #
