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How do you find the minimum value of a quadratic function?
For example:
In this lesson you will learn to rewrite a quadratic function to reveal the minimum value by
completing the square.
Let’s ReviewLet’s Review
Suppose we have
is positive
Let’s ReviewLet’s Review
Complete or exact square
(x+a)2
(x+a)(x+a)
X2+2ax+a2
Let’s ReviewA Common Mistake
Forgetting to preserve equality
=(x2-10x ) +32
=(x-5)2+32
y=(x-5)2 + 7
Add and subtract the same number to keep the equality
Let’s ReviewCore Lesson
y=x2-12x+49
=(x2-12x )+49
=(x-6)2
square - always positive unless it’s zero
Let’s ReviewCore Lesson
ADDING a positive number to any
number makes that number bigger so the function will have a minimum value when the square term is zero.
Let’s ReviewCore Lesson
y
=(6-6)2+13
smallest number
Let’s ReviewCore Lesson x (x-6)2 +13 y
(1-6)2+13
(6-6)2+13
(7-6)2+13
(0-6)2+13The minimum value
is 13 when x=6 y=(x-6)2 + 13
7 14
In this lesson you have learned to rewrite a quadratic function to
reveal the minimum value by completing the square
Let’s ReviewGuided Practice
Rewrite y=x2+8x+3 by completing the square.
What is the minimum value of this quadratic function?
Let’s ReviewExtension Activities
Partner matching cards. Write each polynomial and ordered pair on index cards. Shuffle all 12 cards then match each function with its equivalent form and corresponding minimum value.
y=x2+16x+64 y=(x+8)2 (-8,0)y=x2-2x-9 y=(x-1)2-10 (1,-10)y=x2+26x+68 y=(x+13)2-101 (-13,-101y=x2+18x-4 y=(x+9)2-85 (-9/-85)
Let’s ReviewExtension Activities
Write a quadratic function whose graph has the given characteristics
1. Minimum: (6,1) point on the graph: (4,5)
2. Points on the graph: (1,7), (4,-2), (5,-1)
3. X-intercepts (5,0) (-4,0)
Let’s ReviewQuick Quiz
Rewrite each function to find its minimum value by completing the square. 1. y=x2-3x+9
2. y=x2+12x+24