How do you write the vertex form equation of the parabola y = x^2 - 4x + 3?

1 Answer
Oct 17, 2017

y=(x-color(red)2)^2+color(blue)((-1))
with vertex at (color(red)2,color(blue)(-1))

Explanation:

The general vertex form is
color(white)("XXX")y=(x-color(red)a)^2+color(blue)b
with vertex at (color(red)a,color(blue)b)

+----------------------------------------------------------------------------------+
| The process used (below) is often referred to ascolor(white)("XXXxx") |
| color(white)("XXX")completing the squarecolor(white)("XXXXXXXXXXXXXX..x")|
+----------------------------------------------------------------------------------+

In order to convert the given equation: y=x^2-4x+3
into vertex form, we need to a term of the form (x-color(red)a)^2

Since (x-color(red)a)^2=(x^2+2color(red)ax+color(red)a^2)
and
since the coefficient of x in the given equation is (-4)
then
color(white)("XXX")2color(red)a=-4color(white)("xxx")rarrcolor(white)("xxx")color(red)a=-2
and
color(white)("XXXXXXXXXXXXXXX")color(red)a^2=color(green)4

That is the first term of the vertex form must be
color(white)("XXX")(x-color(red)2)^2

We need to add color(green)4 to x^2-4x from the original equation to make (x-color(red)2)^2
...but if we are going to add color(green)4 then we will also need to subtract color(green)4 so the equation will not really change.

y=x^2-4x+3
color(white)("XXX")will therefore become
y=underbrace(x^2-4xcolor(green)(+4))+underbrace(3color(green)(-4))
color(white)("XXX")re-writing as a squared binomial and simplifying the constants
y=underbrace((x-color(red)2)^2)+underbrace(color(blue)((-1)))
color(white)("XXX")which is the vertex form