Question

How do you find the intercept and vertex of `y= (2x-1)(3x+4)`?

Answer 1

Analyse the equation to find `y` intercept `(0, -4)`, `x` intercepts `(-4/3, 0)`, `(1/2, 0)` and vertex `(-5/12, -121/24)`

Explanation:

If `y=0` then `(2x-1) = 0` or `(3x+4) = 0`

Hence `x=1/2` or `x=-4/3`

If `x=0` then `y = (0-1)(0+4) = -4`

So the `x` intercepts are `(-4/3, 0)` and `(1/2, 0)`

and the `y` intercept is `(0, -4)`

Since parabolas are bilaterally symmetric, the vertex will have `x` coordinate exactly midway between the two `x` intercepts:

`x = (-4/3+1/2)/2 = (-8/6+3/6)/2 = -5/12`

Substitute this value of `x` back into the original equation to find:

`y = (2(-5/12)-1)(3(-5/12)+4)`

`=(-5/6-6/6)(-15/12+48/12)`

`=(-11/6)(33/12)`

`=(-11/6)(11/4)`

`=-121/24`

So the vertex is at `(-5/12, -121/24)`

graph{(2x-1)(3x+4) [-6.58, 5.904, -5.27, 0.97]}