What is the vertex form of #y=(3x – 4) (2x – 1) #?

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Answer 1 · 2018-05-12T21:36:49

#y=6(x-11/12)^2-25/24#

In vertex form, a is stretch factor, h is the x-coordinate of the vertex and k is the y-coordinate of the vertex.

#y=a(x-h)^2+k#

So, we must find the vertex.

The zero product property says that, if #a*b=0#, then #a=0# or #b=0#, or #a,b=0#.

Apply the zero product property to find the roots of the equation.

#color(red)((3x-4)=0)#

#color(red)(3x=4)#

#color(red)(x_1=4/3)#

#color(blue)((2x-1)=0)#

#color(blue)(2x=1)#

#color(blue)(x_2=1/2)#

Then, find the midpoint of the roots to find the x-value of the vertex. Where #M="midpoint"#:

#M=(x_1+x_2)/2#

#" "=(4/3+1/2)/2#

#" "=11/12#

#:. h=11/12#

We can input this value for x in the equation to solve for y.

#y=(3x-4)(2x-1)#

#y=[3(11/12)-4][2(11/12)-1]#

#y=-25/24#

#:. k=-25/24#

Input these values respectively into a vertex-form equation.

#y=a(x-11/12)^2-25/24#

Solve for the a value by inputting a known value along the parabola, for this example, we'll use a root.

#0=a[(1/2)-11/12]^2-25/24#

#25/24=a((-5)/12)^2#

#25/24=25/144a#

#a=6#

#:. y=6(x-11/12)^2-25/24#