Consider the function: f(x) = 2|x – 2| + 1

1.Does the graph of the function open up or down?

2.Is the graph of the function wider, narrower, or the same width as y = |x|?

3.What is the vertex of the graph?

4.What is the line of symmetry for the graph?

5.What numbers would you put in the table to complete the graph?

Algebra II 1

I

Quadratic Functions

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…

3

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)

4

y = a(x – h)2 + kVertex: (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 5

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

26

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

117

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

-28

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

99

Graph:y = -2(x + 2)2

Opens down

Narrower than x2

Vertex: (-2,0)

AOS: x = -2

Table

Reflect

x

-1

0

y

-2

-8I 10

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 11

Graph:y = x2 – 2x – 5

Opens up, Same width as x2

AOS: x = - b / (2a) x = 2 / (21) x = 1

y = (1)2 – 2(1) – 5Vertex: (1, -6)

Table

Reflect

x

2

3

y

-5

-2I 12

Graph:y = 2x2 – 4x + 3

Opens up, Narrower than x2

AOS: x = - b / (2a) x = 4 / (22) x = 1

y = 2(1)2 – 4(1) + 3Vertex: (1, 1)

Table

Reflect

x

2

3

y

3

9I 13

Graph:y = ½x2 + x – 6

Opens up, Wider than x2

AOS: x = - b / (2a) x = -1 / (2½) x = -1

y = ½(-1)2 + (-1) – 6Vertex: (-1, -6.5)

Table

Reflect

x

0

1

y

-6

-4.514

Graph:y = -2x2 + 1

Opens down, Narrower than x2

AOS: x = - b / (2a) x = -0 / (2-2) x = 0

y = -2(0)2 + 1Vertex: (0, 1)

Table

Reflect

x

1

2

y

-1

-715

Graph:y = 2x2 – 4x – 1

Opens up, Narrower than x2

AOS: x = - b / (2a) x = 4 / (22) x = 1

y = 2(1)2 – 4(1) – 1Vertex: (1, -3)

Table

Reflect

x

2

3

y

-1

5I 16

y = a(x – p)(x – q)x-intercepts: (opposite of p, 0) ,

(opposite of q, 0)AOS: x = (opp p + opp q)

2Vertex: (opp p + opp q , f(opp p + opp q) )

2 2

***Do not have to make a table***

17

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)

18

Graph:y = ½(x – 6)(x –

4)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)

19

Graph:y = -½x(x – 5)

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)

20

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 21

Graph:y = ⅓(x – 2)(x +

4)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)

22

23Algebra II

General Strategy for Problem Solving1. 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.

24

25

The equation for the percent of test subjects that felt comfortable at a given temperature x is

y = –3.678x2 + 527.3x – 18,807. What temperature made the greatest percent of test subjects comfortable?

At that temperature, what percent of people felt comfortable?

26

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)2 + 8. What is the distance between the two towers? What is the height of

the cable above the road at its lowest point?

The archway that forms the ceiling of a tunnel can be modeled by the equation y = –0.0355x2 + .923x + 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?

27

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

28II

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?

29

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

30

Find two numbers whose sum is 49 and whose product is a maximum.

31Algebra II

Find two numbers whose product is a maximum if the sum of the first and five times the second is 80.

32Algebra II

Write the standard form of the equation of the parabola whose vertex is (1,2) and passes through (3, -6)

33Algebra II

Write the standard form of the equation of the parabola whose vertex is (-4,11) and that passes through the point (-6,15)

34Algebra II

1. y = -½(x – 3)(x + 1)

2. y = 2(x + 3)2 – 2

3. y = –x2 + 4x – 2

35Algebra II