Question

How do you find the vertex and intercepts for `y = (x-5)^2 +2`?

Answer 1

(5,2) , (0,27) , no x-intercepts

Explanation:

The standard vertex form of a quadratic function is

`color(red)(|bar(ul(color(white)(a/a)color(black)( y =a (x - h)^2 + k)color(white)(a/a)|)))`
where (h , k) are the coordinates of the vertex , and a is a constant.

the function `y = (x - 5)^2 + 2 " is in this form "`

by comparison, the coords of vertex = (5 , 2)

To find where it crosses the y-axis , let x = 0 in the equation.

x = 0 : y = `(-5)^2 + 2 = 25 + 2 = 27 rArr (0 , 27)`

To find where it crosses the x-axis let y = 0

y `= 0 : (x-5)^2 + 2 = 0`

hence `(x-5)^2 = -2 rArr x-5 = ±sqrt-2`

`rArr x = 5 ± 2i " no real solution thus no x-intercepts "`
graph{(x-5)^2+2 [-40, 40, -20, 20]}