Question

What is the equation of the following graph in vertex form?

Answer 1

The equation in vertex form is:

`y=(x-2)^2+1`

Explanation:

Recall that the formula of a parabola with vertex at
`(x_(vertex),y_(vertex))` is given by the equation in vertex form:

`y=a\,(x-x_(vertex))^2+y_(vertex)`

where the parameter
can be specified by an extra information on any other point apart from the vertex, that parabola goes through.

In our case, since the vertex must be the point (2, 1), the vertex form of the parabola becomes:

`y=a\,(x-x_(vertex))^2+y_(vertex)\\y=a\,(x-2)^2+1`

we have the information on the extra point (0, 5) where the parabola crosses the y-axis. Then, we use it to find the missing parameter
`a`:

`y=a\,(x-2)^2+1\\5=a(0-2)^2+1\\5=a\,*\,4+1\\5-1=4\,a\\4=4\,a\\a=1`

The, the final form of the parabola's equation in vertex form is:

`y=(x-2)^2+1`