This document discusses quadratic functions and their maxima and minima. It provides the standard form and vertex form of quadratic equations, and explains how to find the vertex and y-intercept of a quadratic function. It derives the vertex formula, stating that the maximum or minimum of a quadratic occurs at the x-coordinate of the vertex. Finally, it provides homework problems involving finding maxima, minima, vertices, and y-intercepts of various quadratic functions.

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.

2. Standard Form of a quadratic equation

3. Standard Form of a quadratic equation
2
f (x) = ax + bx + c

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.

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)

18. Put y = ax + bx + c into vertex form
2

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.