1. 2.5 Quadratic Functions;
Maxima & Minima
Psalm 55:22 Cast your burden on the LORD, and he
will sustain you; he will never permit the righteous
to be moved.
4. Standard Form of a quadratic equation
2
f (x) = ax + bx + c
Vertex Form of a quadratic equation
5. Standard Form of a quadratic equation
2
f (x) = ax + bx + c
Vertex Form of a quadratic equation
2
f (x) = a ( x − h ) + k
vertex at ( h, k )
6. Standard Form of a quadratic equation
2
f (x) = ax + bx + c
Vertex Form of a quadratic equation
2
f (x) = a ( x − h ) + k
vertex at ( h, k )
Note: your book is wrong in that it names Vertex Form as Standard Form.
Ignore your book on this point.
7.
8. (-b,c) and (i,j) are both Local Minimums
(f,g) is a Local Maximum
9. Find the coordinates of the vertex and the
y-intercept for f (x) = 3x − 6x + 10
2
10. Find the coordinates of the vertex and the
y-intercept for f (x) = 3x − 6x + 10
2
Algebraic Approach:
11. Find the coordinates of the vertex and the
y-intercept for f (x) = 3x − 6x + 10
2
Algebraic Approach:
f (x) = 3( x 2 − 2x ) + 10
12. Find the coordinates of the vertex and the
y-intercept for f (x) = 3x − 6x + 10
2
Algebraic Approach:
f (x) = 3( x 2 − 2x ) + 10
f (x) = 3( x − 2x + 1) + 10 − 3
2
13. Find the coordinates of the vertex and the
y-intercept for f (x) = 3x − 6x + 10
2
Algebraic Approach:
f (x) = 3( x 2 − 2x ) + 10
f (x) = 3( x − 2x + 1) + 10 − 3
2
2
f (x) = 3( x − 1) + 7
vertex : (1, 7 )
14. Find the coordinates of the vertex and the
y-intercept for f (x) = 3x − 6x + 10
2
Algebraic Approach:
f (x) = 3( x 2 − 2x ) + 10
f (x) = 3( x − 2x + 1) + 10 − 3
2
2
f (x) = 3( x − 1) + 7
vertex : (1, 7 )
f (0) = 10
y − intercept : ( 0,10 )
15. Find the coordinates of the vertex and the
y-intercept for f (x) = 3x − 6x + 10
2
Graphic Approach:
16. Find the coordinates of the vertex and the
y-intercept for f (x) = 3x − 6x + 10
2
Graphic Approach:
2
y1 = 3x − 6x + 10
17. Find the coordinates of the vertex and the
y-intercept for f (x) = 3x − 6x + 10
2
Graphic Approach:
2
y1 = 3x − 6x + 10
Now ... how do we find the x-intercepts?
(discuss and do algebraic vs. graphic)
19. Put y = ax + bx + c into vertex form
2
⎛ 2 b ⎞
y = a ⎜ x + x ⎟ + c
⎝ a ⎠
20. Put y = ax + bx + c into vertex form
2
⎛ 2 b ⎞
y = a ⎜ x + x ⎟ + c
⎝ a ⎠
2 2
⎛ 2 b b ⎞ ab
y = a ⎜ x + x + 2 ⎟ + c − 2
⎝ a 4a ⎠ 4a
21. Put y = ax + bx + c into vertex form
2
⎛ 2 b ⎞
y = a ⎜ x + x ⎟ + c
⎝ a ⎠
2 2
⎛ 2 b b ⎞ ab
y = a ⎜ x + x + 2 ⎟ + c − 2
⎝ a 4a ⎠ 4a
2 2
⎛ b ⎞ ⎛ b ⎞
y = a ⎜ x + ⎟ + ⎜ c − ⎟
⎝ 2a ⎠ ⎝ 4a ⎠
22. Put y = ax + bx + c into vertex form
2
⎛ 2 b ⎞
y = a ⎜ x + x ⎟ + c
⎝ a ⎠
2 2
⎛ 2 b b ⎞ ab
y = a ⎜ x + x + 2 ⎟ + c − 2
⎝ a 4a ⎠ 4a
2 2
⎛ b ⎞ ⎛ b ⎞
y = a ⎜ x + ⎟ + ⎜ c − ⎟
⎝ 2a ⎠ ⎝ 4a ⎠
2
y = a(x - h) + k
23. Put y = ax + bx + c into vertex form
2
⎛ 2 b ⎞
y = a ⎜ x + x ⎟ + c
⎝ a ⎠
2 2
⎛ 2 b b ⎞ ab
y = a ⎜ x + x + 2 ⎟ + c − 2
⎝ a 4a ⎠ 4a
2 2
⎛ b ⎞ ⎛ b ⎞
y = a ⎜ x + ⎟ + ⎜ c − ⎟
⎝ 2a ⎠ ⎝ 4a ⎠
2
2 ⎛ −b b ⎞
y = a(x - h) + k vertex: ⎜ 2a , c − 4a ⎟
⎝ ⎠
24. ⎛ −b b 2 ⎞
If the vertex is then the extrema
⎜ 2a , c − 4a ⎟
⎝ ⎠
point (max or min) occurs at:
25. ⎛ −b b 2 ⎞
If the vertex is then the extrema
⎜ 2a , c − 4a ⎟
⎝ ⎠
point (max or min) occurs at:
⎛ −b ⎛ −b ⎞ ⎞
⎜ 2a , f ⎜ 2a ⎟ ⎟
⎝ ⎝ ⎠ ⎠
26. ⎛ −b b 2 ⎞
If the vertex is then the extrema
⎜ 2a , c − 4a ⎟
⎝ ⎠
point (max or min) occurs at:
⎛ −b ⎛ −b ⎞ ⎞
⎜ 2a , f ⎜ 2a ⎟ ⎟
⎝ ⎝ ⎠ ⎠
This is called the Vertex Formula for
quadratics (not to be confused with the
Vertex Form of a quadratic).
27. Pages 201, 202:
Do # 60, 62, 64 algebraically & verify graphically
(do as many as a class as time permits)
HW #6
“Nothing pains some people more than having
to think.” Martin Luther King Jr.