Question

Let f(x)=x^2-4x+3
(a) Put f(x) in standard form f(x)=a(x-h)^2+k
(b) Does f(x) have a maximum or minimum? find it
(please explain)

Answer 1

do you know how to complete the square?

Answer 2

i dont remember how to complete the square

Answer 3

start with

f(x)=x^2-4x+3

then I'm just going to write a set of parenthese around the first two terms

(x^2-4x) + 3

now, we take the (-4) in front of the x

divide that by 2 to get (-2)

and square it to get 4

with me so far?

Answer 4

yes

Answer 5

good, now we add the + 4 inside the parentheses

to make things equal, we must also subtract 4 from the outside

now we have

(x^2 - 4x + 4) + 3 - 4

with me so far?

Answer 6

yep

Answer 7

now factor what's inside the parentheses

let me know if you need help

Answer 8

is it (x+2)(x-2)

Answer 9

close but not quite

one of the signs is wrong

Answer 10

is it (x-2)(x-2)

Answer 11

yup! we can also write that as

(x-2)^2

so

(x^2 - 4x + 4) + 3 - 4

becomes

(x-2)^2 - 1

and that's our standard form for part a

any questions?

Answer 12

nope no questions thank you :) can you help me with part b?

Answer 13

yeah sure

our function has a minimum since there's no negative sign in front of the (x-h)^2 part

Answer 14

to find the minimum, note that:


a(x-h)^2+k

has the minimum (h,k)

so if our function is (x-2)^2 - 1

what is the minimum?

Answer 15

(2,-1)?

Answer 16

good! and that's it for part b

any questions?

Answer 17

no questions thank you sooo much