1. Consider the function: f(x) = 2|x – 2| + 1
Bellwork:
Consider the function: f(x) = 2|x – 2| + 1
Does the graph of the function open up or down?
Is the graph of the function wider, narrower, or the same width as y = |x|?
What is the vertex of the graph?
What is the line of symmetry for the graph?
What numbers would you put in the table to complete the graph?
Algebra II

2. 2.1
Quadratic Functions I

3. Quadratics
A quadratic equation has a squared term in it, or a degree of two.
The graph of a quadratic makes a “U” shape called a parabola.
There are 3 forms a quadratic equation can be in…

4. Three Forms…
1. Vertex Form: y = a(x – h)2 + k 2. Standard Form: y = ax2 + bx + c 3. Intercept Form: y = a(x – p)(x – q)

5. Graphing from vertex form
y = a(x – h)2 + k
Vertex: (opposite of h, same as k)
Axis of symmetry (AOS): x = opposite of h.
a: determines the direction the graph opens, and the width of the graph
a > 0 opens up
a < 0 opens down
|a| < 1 wider than x2
|a| > 1 narrower than x2
|a| = 1 same width as x2 II

6. Example 1 Graph: y = - ½(x + 3)2 + 4 Opens down Wider than x2
Vertex: (-3,4)
AOS: x = -3
Table 
Reflect x -2 -1 y
3.5 2

7. Example 2 Graph: y = 2(x – 1)2 + 3 Opens up Narrower than x2
Vertex: (1,3)
AOS: x = 1
Table 
Reflect x 2 3 y 5 11

8. Example 3 Graph: y = -(x + 5)2 + 2 Opens down Same width as x2
Vertex: (-5,2)
AOS: x = -5
Table 
Reflect x -4 -3 y 1 -2

9. Example 4 Graph: y = 2(x – 1)2 + 1 Opens up Narrower than x2
Vertex: (1,1)
AOS: x = 1
Table 
Reflect x 2 3 y 3 9

10. Example 5 Graph: y = -2(x + 2)2 Opens down Narrower than x2
Vertex: (-2,0)
AOS: x = -2
Table 
Reflect x -1 y -2 -8 I

11. Graphing from standard form
y = ax2 + bx + c
AOS: x = – b/(2a)
Vertex: ( – b/(2a), f(-b/2a))
“a” still determines the direction the graph opens and the width of the parabola
Algebra II

12. Example 6 Graph: y = x2 – 2x – 5 AOS: x = - b / (2a) x = 2 / (21)
Opens up, Same width as x2
AOS: x = - b / (2a)
x = 2 / (21)
x = 1
y = (1)2 – 2(1) – 5
Vertex: (1, -6)
Table 
Reflect x 2 3 y -5 -2 I

13. Opens up, Narrower than x2
Example 7
Graph:
y = 2x2 – 4x + 3
Opens up, Narrower than x2
AOS: x = - b / (2a)
x = 4 / (22)
x = 1
y = 2(1)2 – 4(1) + 3
Vertex: (1, 1)
Table 
Reflect x 2 3 y 3 9 I

14. Example 8 Graph: y = ½x2 + x – 6 AOS: x = - b / (2a) x = -1 / (2½)
Opens up, Wider than x2
AOS: x = - b / (2a)
x = -1 / (2½)
x = -1
y = ½(-1)2 + (-1) – 6
Vertex: (-1, -6.5)
Table 
Reflect x 1 y -6
-4.5

15. Opens down, Narrower than x2
Example 9
Graph:
y = -2x2 + 1
Opens down, Narrower than x2
AOS: x = - b / (2a)
x = -0 / (2-2)
x = 0
y = -2(0)2 + 1
Vertex: (0, 1)
Table 
Reflect x 1 2 y -1 -7

16. Opens up, Narrower than x2
Example 10
Graph:
y = 2x2 – 4x – 1
Opens up, Narrower than x2
AOS: x = - b / (2a)
x = 4 / (22)
x = 1
y = 2(1)2 – 4(1) – 1
Vertex: (1, -3)
Table 
Reflect x 2 3 y -1 5 I

17. Graphing from intercept form
y = a(x – p)(x – q)
x-intercepts: (opposite of p, 0) ,
(opposite of q, 0)
AOS: x = (opp p + opp q) 2
Vertex: (opp p + opp q , f(opp p + opp q) )
***Do not have to make a table***

18. Opens down, Same width as x2
Example 11
Graph:
y = -(x + 2)(x – 4)
Opens down, Same width as x2
Intercepts:
(-2, 0), (4, 0)
AOS: x = (- p + - q) / 2
x = (-2 + 4) / 2
x = 1
y = -(1 + 2)(1 – 4)
Vertex: (1, 9)

19. Example 12 Graph: y = ½(x – 6)(x – 4) Intercepts: (6, 0), (4, 0)
Opens up, Wider than x2
Intercepts:
(6, 0), (4, 0)
AOS: x = (- p + - q) / 2
x = (6 + 4) / 2
x = 5
y = ½(5 – 6)(5 – 4)
Vertex: (5, -½)

20. Example 13 Graph: y = -½x(x – 5) Intercepts: (0, 0), (5, 0)
Opens down, Wider than x2
Intercepts:
(0, 0), (5, 0)
AOS: x = (- p + - q) / 2
x = (0 + 5) / 2
x = 2.5
y = -½(2.5)(2.5 – 5)
Vertex: (2.5, 3.125)

21. Opens down, Narrower than x2
Example 14
Graph:
y = -2(x – 3)(x + 1)
Opens down, Narrower than x2
Intercepts:
(3, 0), (-1, 0)
AOS: x = (- p + - q) / 2
x = (3 + -1) / 2
x = 1
y = -2(1 – 3)(1 + 1)
Vertex: (1, 8) I

22. Example 15 Graph: y = ⅓(x – 2)(x + 4) Intercepts: (2, 0), (-4, 0)
Opens up, Wider than x2
Intercepts:
(2, 0), (-4, 0)
AOS: x = (- p + - q) / 2
x = (2 + -4) / 2
x = -1
y = ⅓(-1 – 2)(-1 + 4)
Vertex: (-1, -3)

23. Algebra II

24. Strategy for Problem Solving
General Strategy for Problem Solving
1. UNDERSTAND the problem.
Read and reread the problem
Choose a variable to represent the unknown
Construct a drawing, whenever possible
2. MODEL the problem with an equation.
3. SOLVE the equation.
4. INTERPRET the result.
Check proposed solution in original problem.
State your conclusion.

25. Example 16
The equation for the percent of test subjects that felt comfortable at a given temperature x is y = –3.678x x – 18,807. What temperature made the greatest percent of test subjects comfortable? At that temperature, what percent of people felt comfortable?

26. Example 17
The Golden Gate Bridge in San Francisco has two towers that rise 500 feet above the road and are connected by cables as shown. Each cable forms a parabola with the equation y = 1/8960(x – 2100) What is the distance between the two towers? What is the height of the cable above the road at its lowest point?

27. Example 18
The archway that forms the ceiling of a tunnel can be modeled by the equation y = –0.0355x x + 10 where x is the horizontal distance in feet and y is the height in feet from the ceiling to the floor. How many feet from the walls does the ceiling reach its maximum height? What is the maximum height?

28. Example 19
The length of a rectangle is three more than twice the width. Determine the dimensions that will give a total area of 27 m2. II

29. Example 20
The length of a Ping-Pong table is 3 ft more than twice the width. The area of a Ping-Pong table is 90 square feet. What are the dimensions of a Ping-Pong table?

30. Example 21
Find two positive whose sum is 32 and whose product is a maximum.

31. Example 22
Find two numbers whose sum is 49 and whose product is a maximum.
Algebra II

32. Example 23
Find two numbers whose product is a maximum if the sum of the first and five times the second is 80.
Algebra II

33. Example 24
Write the standard form of the equation of the parabola whose vertex is (1,2) and passes through (3, -6)
Algebra II

34. Example 25
Write the standard form of the equation of the parabola whose vertex is (-4,11) and that passes through the point (-6,15)
Algebra II

35. Closure: Graph each.
1. y = -½(x – 3)(x + 1) 2. y = 2(x + 3)2 – 2 3. y = –x2 + 4x – 2
Algebra II