18. What is to be learned?
β’ How to complete the square
β’ How to use this to find maximum and
minimum values
19. Minimum Values
y = (x + 3)2
Lowest Value of y?
Try x = 0
y = (0 + 3)2
= 32
= 9
20. Minimum Values
y = (x + 3)2
Lowest Value of y?
Try x = -3
y = (-3 + 3)2
= 02
= 0
Minimum Value of y = 0,
when x = -3
corresponding
value of x
21. (x β 4)2
+ 7
Min Value?
Try x = 4
(4 β 4)2
+ 7
= 02
+ 7
= 7
Min Value 7,
Always get bracket bit equal to zero
when x = 4
22. Maximum Values
12 β (x β 5)2
Try x = 5
12 β 02
= 12
Maximum Value = 12, when x = 5
Often written in form β(x β 5)2
+ 12
Lowest value is zero
23. Putting it all together
Minimum value of
x2
+ 12x + 47
= (x + 6)2
β 36 + 47
= (x + 6)2
+ 11
Minimum value occurs when x =
Minimum Value is
Always get bracket
bit equal to zeroο
ο
-6
11
24. Max and Min Values
Completing the square puts expressions in
form where it is easy to find max or min
value
Tactic is always to make bracket bit = 0
Ex y = x2
+ 10x + 7
= (x + 5)2
β 25 + 7
= (x + 5)2
β 18
so min value occurs when x = -5
y = -18
25. Key Question
Find the minimum value of expression and
corresponding value of x
y = x2
+ 10x + 7
= (x + 5)2
β 25 + 7
= (x + 5)2
β 18
so min value = -18
when x = -5