Question

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

Answer 1

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

Explanation:

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`