1. Objective The student will be able to:
find the slope of a line given 2 points and a graph.
MFCR- Lesson 2-2

2. 10-13-14 Bellwork – formative #14
1. Find the missing coordinate so the ordered pairs are solutions to: ½ x + y = 4
a. (0, ?)
b. (?, 0)
c. (-4, ?)
2. Find the x and y intercepts of 2x+8=0, then graph.
3. Find the x and y intercepts of 2x-2y-6=0, then graph.

3. What is the meaning of this sign?
Icy Road Ahead
Steep Road Ahead
Curvy Road Ahead
Trucks Entering Highway Ahead

4. Slope is the steepness of a line.
What does the 7% mean? 7%
7% is the slope of the road. It means the road drops 7 feet vertically for every 100 feet horizontally.
7 feet
100 feet
So, what is slope???
Slope is the steepness of a line.

5. Slope can be expressed different ways:
A line has a positive slope if it is going uphill from left to right.
A line has a negative slope if it is
going downhill from left to right.

6. 1) Determine the slope of the line.
When given the graph, it is easier to apply “rise over run”.

7. Determine the slope of the line.
Start with the lower point and count how much you rise and run to get to the other point!
rise 3 = =
run 6 6 3
Notice the slope is positive AND the line increases!

8. 2) Find the slope of the line that passes through the points (-2, -2) and (4, 1).
When given points, it is easier to use the formula!
y2 is the y coordinate of the 2nd ordered pair (y2 = 1)
y1 is the y coordinate of the 1st ordered pair (y1 = -2)

9. You can do the problems either way! Which one do you think is easiest?
Did you notice that Example #1 and Example #2 were the same problem written differently? 6 3
(-2, -2) and (4, 1)
You can do the problems either way!
Which one do you think is easiest?

10. Find the slope of the line that passes through (3, 5) and (-1, 4).-4
- ¼

11. 3) Find the slope of the line that goes through the points (-5, 3) and (2, 1).

12. Determine the slope of the line shown.-2 -½ 2

13. Determine the slope of the line.-1
Find points on the graph.
Use two of them and apply rise over run. 2
The line is decreasing (slope is negative).

14. What is the slope of a horizontal line?
The line doesn’t rise!
All horizontal lines have a slope of 0.

15. What is the slope of a vertical line?
The line doesn’t run!
All vertical lines have an undefined slope.

16. Remember the word “VUXHOY”
Vertical lines
Undefined slope
X = number; This is the equation of the line.
Horizontal lines
O - zero is the slope
Y = number; This is the equation of the line.

17. Find the Slope of the Line
Given the points (5, 1) & (6, 4)
m =
= = 3
Given the points (-6, -2) & (-1, 0)
m =

18. Exit Ticket – Find the slope of each line. #1 and #2

19. Bellwork
Find the slope of the line that passes through the given points.
(1,-8) and (-5, -4)
(5, -2) and (5, 5)
(1,6) and (-7,6)

20. Bellwork solutions
-2/3
Undefined

21. Lesson 2-2 continued Parallel Slopes

22. Perpendicular Slopes

23. Parallel Lines What do you know about the slope of parallel lines?
Parallel lines have the SAME SLOPE!
What is the slope of the line parallel to y = -2x + 4?
m = -2

24. Perpendicular Lines
What do you know about the slope of perpendicular lines?
Perpendicular lines have slopes that are opposite reciprocals!
What is the slope of the line perpendicular to y = -2x + 4?
m = 1/2

25. Summary Parallel slopes are the same.
Perpendicular slopes are opposite reciprocals.

26. Classify as parallel or perpendicular
1. Line 1: (4, -1), (-3, 6) Line 2: (-1,3), (2,0) 2. Line 1: (0,0), (2,3) Line 2: (-2, 5), (0,-2) 3. Line 1: (2, 5), (4, 9) Line 2: (-1, 4), (3, 2)

27. CW: Int. Algebra Book
Pages # 8-54 evens

28. Bellwork
What is the slope of the line perpendicular to y = -2x + 4?
What is the slope of the line parallel to
y = -2x + 4?

29. Friday’s CW solutions 2.2 slope worksheets

30. Lesson 2-3 Objective The student will be able to:
write equations using slope-intercept form.
identify slope and y-intercept from an equation

31. b represents the y-intercept
Important!!!
This is one of the big concepts in Algebra 1. You need to thoroughly understand this!
Slope – Intercept Form
y = mx + b
m represents the slope
b represents the y-intercept

32. Writing Equations
When asked to write an equation, you need to know two things – slope (m) and y-intercept (b).
There are three types of problems you will face.

33. Writing Equations – Type #1
Write an equation in slope-intercept form of the line that has a slope of 2 and a y-intercept of 6.
To write an equation, you need two things:
slope (m) =
y – intercept (b) =
We have both!! Plug them into slope-intercept form
y = mx + b
y = 2x + 6 2 6

34. Write the equation of a line that has a y-intercept of -3 and a slope of -4.
y = -3x – 4
y = -4x – 3
y = -3x + 4
y = -4x + 3

35. Writing Equations – Type #2
Write an equation of the line that has a slope of 3 and goes through the point (2,1).
To write an equation, you need two things:
slope (m) =
y – intercept (b) =
We have to find the y-intercept!! Plug in the slope and ordered pair into y = mx + b
1 = 3(2) + b 3
???

36. Writing Equations – Type #2
1 = 3(2) + b
Solve the equation for b
1 = 6 + b
-5 = b
To write an equation, you need two things:
slope (m) =
y – intercept (b) =
y = 3x - 5 3 -5

37. Writing Equations – Type #3
Write an equation of the line that goes through the points (-2, 1) and (4, 2).
To write an equation, you need two things:
slope (m) =
y – intercept (b) =
We need both!! First, we have to find the slope. Plug the points into the slope formula.
Simplify
???
???

38. Writing Equations – Type #3
Write an equation of the line that goes through the points (-2, 1) and (4, 2).
To write an equation, you need two things:
slope (m) =
y – intercept (b) =
It’s now a Type #2 problem. Pick one of the ordered pairs to plug into the equation. Which one looks easiest to use?
I’m using (4, 2) because both numbers are positive.
2 = (4) + b
???

39. Writing Equations – Type #3
2 = (4) + b
Solve the equation for b
2 = b
To write an equation, you need two things:
slope (m) =
y – intercept (b) =

40. Write an equation of the line that goes through the points (0, 1) and (1, 4).
y = 3x + 4
y = 3x + 1
y = -3x + 4
y = -3x + 1

41. To find the slope and y-intercept of an equation, write the equation in slope-intercept form: y = mx + b.
Find the slope and y-intercept.
y = 3x – 7
y = mx + b
m = 3, b = -7

42. Find the slope and y-intercept.
2) y = x
y = mx + b
y = x + 0
3) y = 5
y = 0x + 5
m =
b = 0
m = 0
b = 5

43. Find the slope and y-intercept. 4) 5x - 3y = 6
Write it in slope-intercept form. (y = mx + b)
5x – 3y = 6
-3y = -5x + 6
y = x - 2 -3 -3 -3
m =
b = -2

44. Find the slope and y-intercept. 5) 2y + 2 = 4x
Write it in slope-intercept form. (y = mx + b)
2y + 2 = 4x
2y = 4x - 2
y = 2x - 1 2 2 2
m = 2
b = -1

45. Find the slope and y-intercept of y = -2x + 4
m = 2; b = 4
m = 4; b = 2
m = -2; b = 4
m = 4; b = -2

46. Graphing a Line Given a Point & Slope
Graph a line though the point
(2, -6) with m=2/3
Graph (2, -6)
Count up 2 for the
rise, and to the right
3 for the run
Plot the point, repeat, then connect x y

47. Graphing Lines m = - ½ b = 3 Plot y-intercept (b) Use the slope to
find two more
points
Connect x y

48. Graph the Line
(3, -3) m = undefined x y

49. POINT-SLOPE FORM
Chapter 2-3

50. WarPPPPPpm-Up
BELLWORK – Thurs
Write an equation of the line in slope-intercept form.
1. passes through (3, 4), m = 3
ANSWER
y = 3x – 5
2. passes through (–2, 2) and (1, 8)
ANSWER
y = 2x + 6

51. Warm-Up
3. A carnival charges an entrance fee and a ticket fee.
One person paid $27.50 and brought 5 tickets. Another paid $45.00 and brought 12 tickets. How much will 22 tickets cost?
ANSWER
$70

52. Example 1
Write an equation in point-slope form of the line that passes through the point (4, –3) and has a slope of 2.
Write point-slope form.
y – y1 = m(x – x1)
y + 3 = 2(x – 4)
Substitute 2 for m, 4 for x1, and –3 for y1.

53. Guided Practice
Write an equation in point-slope form of the line that passes through the point (–1, 4) and has a slope of –2. 1.
y – 4 = –2(x + 1)
ANSWER

54. Plot the point (3, –2). Find a second
Example 2
y + 2 = (x – 3). 2 3
Graph the equation
SOLUTION
Because the equation is in point-slope form, you know that the line has a slope of and passes through the point (3, –2). 2 3
Plot the point (3, –2). Find a second
point on the line using the slope.
Draw a line through both points.

55. Guided Practice –
Graph the equation 2.
y – 1 = (x – 2).
ANSWER

56. Example 3
Write an equation in point-slope form of the line shown.
SOLUTION
STEP 1
Find the slope of the line. =
y2 – y1
x2 – x1 m
3 – 1
–1 – 1 2 –2 –1

57. Example 3
STEP 2
Write the equation in point-slope form. You can use either given point.
Method 1
Method 2
Use (–1, 3).
Use (1, 1).
y – y1 = m(x – x1)
y – y1 = m(x – x1)
y – 3 = –(x +1)
y – 1 = –(x – 1)
CHECK
Check that the equations are equivalent by writing them in slope-intercept form.
y – 3 = –x – 1
y – 1 = –x + 1
y = –x + 2
y = –x + 2

58. Guided Practice
Write an equation in point-slope form of the line that passes through the points (2, 3) and (4, 4). 3.
y – 3 = (x – 2) or 1 2
y – 4 = (x – 4)
ANSWER

59. Example 5
WORKING RANCH
The table shows the cost of visiting a working ranch for one day and night for different numbers of people. Can the situation be modeled by a linear equation? Explain. If possible, write an equation that gives the cost as a function of the number of people in the group.

60. Lesson Quiz
Write an equation in point-slope form of the line that passes through (6, –4) and has slope 2. 1.
ANSWER
y + 4 = –2(x – 6)
Write an equation in point-slope form of the line that passes through (–1, –6) and (3, 10). 2.
ANSWER
y + 6 = 4(x + 1) or y –10 = 4(x–3)

61. Lesson Quiz
A travel company offers guided rafting trips for $875 for the first three days and $235 for each additional day. Write an equation that gives the total cost (in dollars) of a rafting trip as a function of the length of the trip. Find the cost for a 7-day trip. 3.
ANSWER
C = 235t + 170, where C is total cost and t is time (in days); $1815

62. Write Equations and Parallel and Perpendicular Lines

63. Warm-Up
Are the lines parallel? Explain.
1. y – 2 = 2x, 2x + y = 7
ANSWER
No; one slope is 2 and the other is –2.
2. –x = y + 4, 3x + 3y = 5
ANSWER
Yes; both slopes are –1.

64. Example 1
Write an equation of the line that passes through (–3, –5) and is parallel to the line y = 3x – 1.
SOLUTION
STEP 1
Identify the slope. The graph of the given equation has a slope of 3. So, the parallel line through (–3, –5) has a slope of 3.

65. Find the y-intercept. Use the slope and the given point.
Example 1
STEP 2
Find the y-intercept. Use the slope and the given point.
y = mx + b
Write slope-intercept form.
–5 = 3(–3) + b
Substitute 3 for m, 3 for x, and 5 for y.
4 = b
Solve for b.
STEP 3
Write an equation. Use y = mx + b.
y = 3x + 4
Substitute 3 for m and 4 for b.

66. Guided Practice
1. Write an equation of the line that passes through
(–2, 11) and is parallel to the line y = –x + 5.
y = –x + 9
ANSWER

67. Example 2
Determine which lines, if any, are parallel or perpendicular.
Line a: y = 5x – 3
Line b: x + 5y = 2
Line c: –10y – 2x = 0
SOLUTION
Find the slopes of the lines.
Line a: The equation is in slope-intercept form. The slope is 5.
Write the equations for lines b and c in slope-intercept form.

68. Example 2
Line b:
x + 5y = 2
Line c:
–10y – 2x = 0
–10y = 2x
5y = – x + 2 x
y = 2 5 1 + –
y = – x 1 5
ANSWER
Lines b and c have slopes of – , so they are
parallel. Line a has a slope of 5, the negative reciprocal
of – , so it is perpendicular to lines b and c. 1 5

69. Guided Practice
Determine which lines, if any, are parallel or perpendicular.
Line a: 2x + 6y = –3
Line b: y = 3x – 8
Line c: –1.5y + 4.5x = 6
ANSWER
parallel: b and c; perpendicular: a and b, a and c

70. Example 3
The Arizona state flag is shown in a coordinate plane. Lines a and b appear to be perpendicular. Are they?
STATE FLAG
Line a: 12y = –7x + 42
Line b: 11y = 16x – 52
SOLUTION
Find the slopes of the lines. Write the equations in slope-intercept form.

71. Example 3
Line a: 12y = –7x + 42
y = – x + 12 42 7
Line b: 11y = 16x – 52 11 52
y = x – 16
ANSWER
The slope of line a is – The slope of line b is
The two slopes are not negative reciprocals, so lines a and b are not perpendicular. 7 12 16 11

72. Guided Practice
3. Is line a perpendicular to line b? Justify your answer using slopes.
Line a: 2y + x = –12
Line b: 2y = 3x – 8
ANSWER
No; the slope of line a is – , the slope of line b is The slopes are not negative reciprocals so the lines are not perpendicular. 1 2 3

73. Example 4
Write an equation of the line that passes through (4, –5) and is perpendicular to the line y = 2x + 3.
SOLUTION
STEP 1
Identify the slope. The graph of the given equation has a slope of 2. Because the slopes of perpendicular lines are negative reciprocals, the slope of the perpendicular line through (4, –5) is . 1 2 –

74. Find the y-intercept. Use the slope and the given point.
Example 4
STEP 2
Find the y-intercept. Use the slope and the
given point.
y =
mx + b
Write slope-intercept form.
–5 =
– (4) + b 1 2
Substitute – for m, 4 for x, and
–5 for y. 1 2
–3 = b
Solve for b.
STEP 3
Write an equation.
y = mx + b
Write slope-intercept form.
y = – x – 3 1 2
Substitute – for m and –3 for b. 1 2

75. Guided Practice
4. Write an equation of the line that passes through (4, 3) and is perpendicular to the line y = 4x – 7.
y = – x + 4 1 4
ANSWER

76. Lesson Quiz 1.
Write an equation of the line that passes through the point (–1, 4) and is parallel to the line y = 5x – 2.
y = 5x + 9
ANSWER
Write an equation of the line that passes through the point (–1, –1) and is perpendicular to the line y = x + 2. 1 4 – 2.
y = 4x + 3
ANSWER

77. Lesson Quiz 3.
Path a, b and c are shown in the coordinate grid. Determine which paths, if any, are parallel or perpendicular. Justify your answer using slopes.
ANSWER
Paths a and b are perpendicular because their slopes, 2
and are negative reciprocals. No paths are parallel. 1 2 –

78. Graphing Lines Graph the line perpendicular to
y = 2x + 3 that goes through the point (-2, 3).
The slope of the line is 2 so the
slope of the perpendicular line is
-1/2.
m = -1/2 b = 3

79. Graphing Lines
Graph the line parallel to x = -1 that goes through the point
(3, -3).
The slope of the line is undefined (VUX) so the slope of the parallel line is undefined.

80. Writing Equations in Standard Form
MFCR Lesson 2-3

81. Warm-Up
Write an equation in point-slope form of the line that passes through the given points.
1. (1, 4), (6, –1)
ANSWER
y – 4 = –(x – 1) or y + 1 = –(x – 6)
2. (
–1, –2), (2, 7)
ANSWER
y + 2 = 3(x + 1) or y – 7 = 3(x – 2)

82. Example 1
Write two equations in standard form that are equivalent to 2x – 6y = 4.
SOLUTION
To write one equivalent equation, multiply each side by 2.
To write another equivalent equation, multiply each side by 0.5.
4x – 12y = 8
x – 3y = 2

83. Write an equation in standard form of the line shown.
Example 2
Write an equation in standard form of the line shown.
SOLUTION
STEP 1
Calculate the slope. –3
m =
1 – (–2)
1 – 2 = 3 –1
STEP 2
Write an equation in point-slope form. Use (1, 1).
y – y1 = m(x – x1)
Write point-slope form.
y – 1 = –3(x – 1)
Substitute 1 for y1, 3 for m and 1 for x1.

84. Rewrite the equation in standard form.
Example 2
STEP 3
Rewrite the equation in standard form.
3x + y = 4
Simplify. Collect variable terms on one side, constants on the other.

85. Guided Practice
Write an equation in standard form of the line through (3, –1) and (2, –3). 2.
–2x + y = –7
ANSWER

86. Example 3
Write an equation of the specified line.
Blue line a.
Red line b.
SOLUTION
The y-coordinate of the given point on the blue line is –4. This means that all points on the line have a y-coordinate of –4. An equation of the line is y = –4. a.
The x-coordinate of the given point on the red line is 4. This means that all points on the line have an x-coordinate of 4. An equation of the line is x = 4. b.

87. Guided Practice
Write equations of the horizontal and vertical lines that pass through the given point.
3. (–8, –9)
y = –9, x = –8
ANSWER
4. (13, –5)
y = –5, x = 13
ANSWER

88. Lesson Quiz
Write an equation in standard form of the line that passes through the given point and has the given slope m or that passes through the two given points. 1.
(1, –6), m = –2
ANSWER
2x + y = –4
2. (–4, –3), (2, 9)
ANSWER
–2x + y = 5