Complex numbers
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2. Complex Numbers. ALGEBRA 2 LESSON 5-6. 1. Simplify – –169. 2. Find |–1 + i | Simplify each expression. (–7 i )(–3 i ) 5. (–3 + 4 i ) – (–3 – 8 i ) 7. Solve x 2 + 2 = 0. –13 i. –21. 12 i. 1 2. ±2 i. Translating Parabolas. ALGEBRA 2 LESSON 5-3.

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Complex Numbers

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Complex numbers

2

Complex Numbers

ALGEBRA 2 LESSON 5-6

  • 1.Simplify – –169.

  • 2.Find |–1 + i|

  • Simplify each expression.

  • (–7i)(–3i)

  • 5.(–3 + 4i) – (–3 – 8i)

  • 7.Solve x2 + 2 = 0.

–13i

–21

12i

1

2

±2i


Translating parabolas

Translating Parabolas

ALGEBRA 2 LESSON 5-3

(For help, go to Lesson2-6.)

Identify the parent function of each function. Then graph the function by translating the parent function.

1.y = –x + 22.y = |3x| + 23.y = –|x + 1| – 1

4.y = 2x, 2 units down5.y = x, 4 units up, 1 unit right

Write an equation for each translation.

5-3


Translating parabolas1

Translating Parabolas

ALGEBRA 2 LESSON 5-3

Solutions

1.y = –x + 2

parent function: y = –x

translate 2 units up

3.y = –|x + 1| – 1

parent function: y = –x

translate 1 unit left and one unit down

2.y = |3x| + 2

parent function: y = |3x|

translate 2 units up

4.y = 2x, 2 units down: y = 2x – 2

5.y = x, 4 units up and 1 unit right: y = (x – 1) + 4, or y = x + 3

5-3


Complex numbers1

25.–5 + 3i

26.–9 – i

27.3 + 2i

28.4 – 7i

29.6 + 3i

30.1 – 7i

31.7 + 4i

32.–2 – 3i

33.10 + 6i

34.–7 – 10i

35.10

36.26 – 7i

37.9 + 58i

38.9 – 23i

39.–36

40.65 + 72i

41.± 5i

42.±

43.±

44.± i 7

45.±6i

46.±

47.–i, –1 – i, i

48.–2i, –4 – 2i, 12 + 14i

49.1 – i, 1 – 3i, –7 – 7i

50.± i 65

51.±7i

8i 3

3

i 15

3

i 2

2

Complex Numbers

ALGEBRA 2 LESSON 5-6

5-6


Complex numbers2

Complex Numbers

ALGEBRA 2 LESSON 5-6

54.a.A: –5, B: 3 + 2i, C: 2 – i, D: 3i, E: –6 – 4i, F: –1 + 5i

b.5, –3 – 2i, –2 + i, –3i, 6 + 4i, 1 – 5i

55.a.Check students’ work.

b.a circle with radius 10 and center at the origin

56.–5, 5

57.288i

58.–1 + 5i

59.10 – 4i

60.8 – 2i

61.11 – 5i

62. 6 + 10i

63.7 – i

64.10 + 11i

65.–27 + 8i

66.–13 + I

5-6


Properties of parabolas

Properties of Parabolas

  • The graph of y = ax2 + bx + c is a parabola when a ≠ 0.

  • When a > 0, the parabola opens up. When a < 0, the parabola opens down.

  • The axis of symmetry is the line x = from completing the square

  • The x-coordinate of the vertex is The y-coordinate of the vertex is the value of y when x = , or y = f( )

  • The y-intercept is (0, c).


Properties of parabolas1

b

2a

2

2(– )

x = – = – = 4Find the x-coordinate of the vertex.

1

4

1

4

y = – (4)2 + 2(4) – 3 = 1Find the y-coordinate of the vertex.

Graph the vertex and the axis of symmetry x = 4.

Graph two points on one side of the axis of symmetry, such as (6, 0) and (8, –3).

Then graph corresponding points (2, 0) and (0, –3).

The maximum value of the function is 1.

Properties of Parabolas

ALGEBRA 2 LESSON 5-2

1

4

Graph y = – x2 + 2x – 3. What is the maximum value of the function?

Since a < 0, the graph of the function opens down, and the vertex represents the maximum value. Find the coordinates of the vertex.

5-2


Properties of parabolas2

Relate:revenue equals price times number of get-away packages sold

Define:Line R = revenue. Let p = price of a get-away package.

Let –0.12p + 60 = number of a get-away packages sold.

Write:R = p ( –0.12p + 60 )

= –0.12p2 + 60pWrite in standard form.

Properties of Parabolas

ALGEBRA 2 LESSON 5-2

The number of weekend get-away packages a hotel can sell is modeled by –0.12p + 60, where p is the price of a get-away package. What price will maximize revenue? What is the maximum revenue?

5-2


Properties of parabolas3

b

2a

60

2(–0.12)

p = – = – = 250Find p at the vertex.

Properties of Parabolas

ALGEBRA 2 LESSON 5-2

(continued)

Find the maximum value of the function. Since a < 0, the graph of the function opens down, and the vertex represents a maximum value.

R = –0.12(250)2 + 60(250)Evaluate R for p = 250

= 7500Simplify.

A price of $250 will maximize revenue. The maximum revenue is $7500.

5-2


Properties of parabolas4

Properties of Parabolas

  • Vertex form of a quadratic function The vertex form of a quadratic function is y = a(x − h)2 + k. The coordinates of the vertex of the parabola are (h, k).


Properties of parabolas5

Properties of Parabolas

  • Each function in the first column is written in standard form, in the vertex form in the second column. Check that each function is equivalent.

  • Patterns Copy and complete the table.

  • Compare the values of and h in each row. Write a formula to show the relationship between and h.


Vertex form

Vertex form

  • The graph of y = a(x − h)2 + k is the graph of y = ax2 translated h units horizontally and k units vertically.

  • When h is positive the graph shifts right; when h is negative the graph shifts left.

  • When k is positive the graph shifts up; when k is negative the graph shifts down.

  • The vertex is (h, k), and the axis of symmetry is the line x = h.


Translating parabolas2

2

3

The graph of y = (x + 1)2 – 2 is a translation of the graph of the parent function y = x2.

2

3

Step 1:Graph the vertex (–1, –2). Draw the axis of symmetry x = –1.

Step 2:Find another point. When x = 2,

y = (2 + 1)2 – 2 = 4. Graph (2, 4).

2

3

Step 3:Graph the point corresponding to (2, 4). It is three units to the left of the axis of symmetry at (–4, 4).

Step 4:Sketch the curve.

Translating Parabolas

ALGEBRA 2 LESSON 5-3

2

3

Graph y = (x + 1)2 – 2.

You can graph it by translating the parent function or by finding the vertex and the axis of symmetry.

5-3


Translating parabolas3

b

2a

x = –

Find the x-coordinate of the vertex.

Substitute for a and b.

(–70)

2(–7)

= –

= –5

y = –7 (–5)2 – 70(–5) – 169

Find the y-coordinate of the vertex.

= 6

y = a(x – h)2 + k

Write in vertex form.

= –7(x – (–5))2 + 6

Substitute for a, h and k.

= –7(x + 5)2 + 6

Translating Parabolas

ALGEBRA 2 LESSON 5-3

Write y = –7x2 – 70x – 169 in vertex form.

The vertex is at (–5, 6).

The vertex form of the function is y = –7(x + 5)2 + 6.

5-3


Assignment 41

Assignment 41

  • Page 244 22, 24, 29

  • Page 251 2,4, 24, 28, 52


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