If f (x) = x2 - 6x + 19, complete the square and determine the minimum or maximum value of the function

minimum value = 10

Step-by-step explanation:

The equation of a parabola in vertex form is

y = a (x - h) ² + k

where (h, k) are the coordinates of the vertex and a is a multiplier

To obtain this form using completing the square

add/subtract (half the coefficient of the x - term) ² to x² - 6x

f (x) = x² + 2 ( - 3) x + 9 - 9 + 19

= (x - 3) ² + 10 ← in vertex form

with (h, k) = (3, 10)

To determine maximum or minimum consider the value of a

• If a > 0 then minimum value of k

• If a < 0 then maximum value of k

Here a = 1 > 0, hence minimum value of k = 10