Describing Quadratic Functions

We are given the function

f (x) = 0.25 (2 x - 15)^{2} + 150,

and we need to determine if it has an absolute maximum and absolute minimum value. Notice it is not given in vertex form, since it contains

2 x

within the square. Let's rewrite it.

f (x) = 0.25 (2 x - 15)^{2} + 150

RewriteRewrite

15

as

2 \cdot 7.5

f (x) = 0.25 (2 x - 2 \cdot 7.5)^{2} + 150

FactorOutFactor out

2

f (x) = 0.25 (2 (x - 7.5))^{2} + 150

(a \cdot b)^{m} = a^{m} \cdot b^{m}

f (x) = 0.25 \cdot 2^{2} (x - 7.5)^{2} + 150

f (x) = (x - 7.5)^{2} + 150

The final equation is written in vertex form

f (x) = a (x - h)^{2} + k,

where

a = 1 .

Since

a > 0,

the parabola opens upward and has a minimum value of

k = 150 .

Describing Quadratic Functions 1.5 - Solution