1. UNIT 2 – Linear Functions
Section 2 – Applications of Linear Functions

2. Parallel and Perpendicular Lines
Supplemental Page 13:
Graph y = ¾ x – 2 and y = ¾ x + 1
a. They have the same slope
b. They are parallel
c. Equations that have the same slope but different y-intercepts are parallel.

3. Parallel and Perpendicular Lines
Graph y = -3x + 5
1. i) A parallel line would have the same slope so m = -3
ii) b= y-intercept. Your new line will have a different y- intercept from your old line. We must find it using y=mx+b
y= mx + b  2 = -3(-4) + b
2= 12 + b  b = -10
iii) y = -3x – 10 This is parallel to the original.

4. Parallel and Perpendicular Lines
Supplemental Page 13:
Graph y = 2x – 3 and y = - ½ x + 1
a. They are opposites and reciprocals
b. They are perpendicular (form 90 degree angle)
c. Equations that have slopes that are opposite reciprocals of each other are perpendicular.

5. Parallel and Perpendicular Lines
Graph y = ¼ x – 1
1. i) A perpendicular line would have the opposite reciprocal slopes so m = -4 for the new line.
ii) b= y-intercept. y= mx + b  -5 = -4(3) + b
-5 = b  b = 7
iii) y = -4x This is perpendicular to the original.

6. Parallel and Perpendicular Lines
Example 2: 4x – 3y = 9
First, what is the slope of this line?
-3y = -4x + 9
y = 4/3 x – 3 m = 4/3
What is the slope for our perpendicular line?
m = -3/4

7. Parallel and Perpendicular Lines
Example 3:
a. y= 5 is a horizontal line.
b. Perpendicular to horizontal is vertical.
Vertical lines are written as x = ____ So x = 8 (from the point)
c. x = -2 is vertical. Perpendicular would be horizontal.
So y = 4 would be perpendicular to x = -2 through that point.

8. Parallel and Perpendicular Lines
PRACTICE:
Supplemental packet page 14

9. Predictions Supplemental packet page 15 Example 1: a. (9, 56) (12, 65)
b. m = 9/3 = 3
c. y = mx + b  56 = 3(9) + b  b = 29
d. y = 3x + 29
means that in 2000 they started with 29,000 people and have been gaining 3,000 people per year.

10. Predictions Example 1: y = 3x + 29 e. Use the TABLE App (7) Y1: 3x+29
i (x = 17) y = 80 (80,000 people)
ii (x = 18) y = 83 (83,000 people)
iii (x = 20) y = 89 (89,000 people)
iv (x = 25) y = 104 (104,000 people)
v (x = 60) y = 209 (209,000 people)
f. Many things might change, too far to predict.

11. Predictions Example 1: y = 3x + 29 g. 115,000 is a y value
So in the year 2028 (about half way through)

12. Predictions with raw data
Example 2:
a. population increases 3 thousand every 2 years.
m = 3/2
b = 3/2 (6) + b (Use first point 6,52)
b = 43
c. y = 3/2 x + 43
means that in 2000 they started with 43,000 people and have been gaining 3,000 people every 2 years.

13. Predictions with raw data
Example 2: y = 3/2 x + 43
d. Use the TABLE App (7) Y1: 3/2 x+43
i (x = 17) y = 68.5 (68,500 people)
ii (x = 18) y = 70 (70,000 people)
iii (x = 20) y = 73 (73,000 people)
iv (x = 25) y = 80.5 (80,500 people)
e. 100 = 3/2 x + 43
x = 38 so 2038

14. Predictions with Scatter Plots
Scatter Plots – A group of pairings that show a general pattern.
1) A)
B) Scatter plot: PRIZM
Use the Statistics App (2)
Enter Year into List 1 and Pop. in List 2
Press F1 Graph and F1 Graph 1
*You may have to use F6 SET to set it up as a scatter plot

15. Predictions with Scatter Plots
C) There are several pairings that will work, we will use the 2nd and last. (5, 42) (22, 80)
D) m = 80 – 42/22 – 5 = 42/17 = about 2.47 (2,470 per year)
42 = 2.47(5) + b  b = 29.65
y = 2.47x
E) y = 2.47(28) = (98,810 people)
200 = 2.47x  x = (2059)

16. Predictions with Scatter Plots
PRACTICE:
Text book p. 81: 12, 13

17. Linear Regression Supplemental Packet p. 17
Practice: Text book p. 87: 2, 3
Assignment: Supplemental Packet p. 18
(y-intercepts exist when x = 0, plug 0 in for x)

18. Graph Absolute Value Functions
Supplemental Packet p y = |x|
a) Vertex is where it changes direction. (0, 0)
b) The pattern is up 1, over 1. i.e. it is using a slope of 1 in both directions.
c) PARENT function: y = |x| uses the points x y

19. Graph Absolute Value Functions
Translations:
a – stretches and shrinks the graph by multiplying the y – values by a.
h – shifts the graph left and right (add to x)
k – shifts the graph up and down (add to y)
y = a |x – h| + k

20. Graph Absolute Value Functions
Example 1: y = 3|x – 2| + 4
a) parent x y
b) a = 3 h = 2 k = 4
c) x (+2)
y(*3 and +4)

21. Graph Absolute Value Functions
Example 1: y = 3|x – 2| + 4
c) x (+2)
y(*3 and +4)
d) vertex is (2, 4)
e) vertex will always be (h, k)
*Note: the pattern in the y values is a slope of 3 in both directions (a =3 is like slope)

22. Graph Absolute Value Functions
Example 2: y = -2|x + 3| + 2
1) vertex is (-3, 2) *(h, k)
2)y-intercept y = -2|0 + 3| + 2
y = -2|3| + 2 = -4 so (0, -4)

23. Graph Absolute Value Functions
Example 2: y = -2|x + 3| + 2
3) x-intercept 0 = -2|x + 3| + 2
-2 = -2|x + 3| *minus 2
1 = |x + 3| * divide by -2
x + 3 = -1 and x + 3 = 1 *absolute value rule
x = -4 and so, (-4, 0) and (-2, 0)

24. Graph Absolute Value Functions
Example 2: y = -2|x + 3| + 2
4) x y
New x *x minus 3
y *y times(-2) plus 2
Notice the slope of -2 in both directions.

25. Graph Absolute Value Functions
PRACTICE – supplemental packet page 20 top of page

26. Linear Inequalities Example 1 (Supplemental packet p. 20, 21)
A) 50x + 30y < 15000
B) x-intercept 50x +30(0) < 15000
x = (300, 0)
y-intercept 50(0) +30y < 15000
y = (0, 500)

27. Linear Inequalities Example 1 A) 50x + 30y < 15000
The inequality is a < so we will shade below the line and use a solid line.
For > we would shade above the line
For > or < we would use a dashed line to indicated that it is not equal to the line.

28. Linear Inequalities Example 1 C) Many points would work. (50, 350)
(200, 100)
(150,350) This means making 150 of type 1 and 350 of type 2 would keep you under cost.

29. Linear Inequalities Example 2 A) 40x – 80y > 8000
B) y > -40x
y < ½ x – 100
*Must reverse the inequality since we divided by a negative 80.

30. Linear Inequalities Example 2 C) F3 Type, F5 Convert
Press F6 graph, then F3 for Window
xmin: -100 xmax:500
ymin:-150 ymax:150

31. Linear Inequalities Example 2 (400, 50)
Means that if they sell 400 products they can have 50 employees working and still make what they need.

32. Linear Inequalities Example 3: a) 9x – 4y < 12 -9x -9x
y > 9/4 x - 3

33. Linear Inequalities Example 3: b) Graph app (5) Y1: 9/4 x – 3
CONVERT to >, notice the dashed line.
c) y-intercept (0, -3)
x-intercept (F5 Gsolv – F1 Root) (-1.3, 0)

34. Linear Inequalities Example 4: 3x – 8y > 16 x-intercept
y-intercept
3(0) – 8y > 16
y < -2 (divide by negative) (0, -2)

35. Linear Inequalities Example 4: 3x – 8y > 16 x-intercept
y-intercept
3(0) – 8y > 16
y < -2 (divide by negative) (0, -2)

36. Linear Inequalities
PRACTICE –supplemental packet page 22