MF10 5 6-Properties Linear Relations

1. Properties of Linear Relations
Slide 1
In a linear relation the graph is a straight line and both the
dependent and independent variables have constant change.
Distance (km) Cost ($)
0 60
100 80
200 100
300 120
400 140
Independent
Variable
Dependent
Variable
+20
+20
+20
+20
+100
+100
+100
+100

2. Properties of Linear Relations
Slide 2
In a linear relation the graph is a straight line and both the
dependent and independent variables have constant change.
0
20
40
60
80
100
120
140
160
0 50 100 150 200 250 300 350 400 450
Cost($)
Distance (km)
Car Rental Cost
+20
Change in
Dependent Variable
+100
Change in
Independent Variable

3. Slide 3
Example 1: Which table of values represents a linear relation?
Justify the answer.
Celsius
(C)
Fahrenheit
(F)
0 31
5 41
10 50
15 59
20 68
The relationship between
temperature in degrees
Celsius, C, and temperature
in degrees Fahrenheit, F.
Amps
(I)
Watts
(P)
0 0
5 75
10 300
15 675
20 1200
The relationship between
the current, I amps, and
power, P watts, in an
electrical current.

4. Slide 4
Example 1: Which table of values represents a linear relation?
Justify the answer.
Celsius
(C)
Fahrenheit
(F)
0 31
5 41
10 50
15 59
20 68
Amps
(I)
Watts
(P)
0 0
5 75
10 300
15 675
20 1200
+5
+5
+5
+5
+9
+9
+9
+9
This is a linear relation.
The changes in both
variables are constant.

5. Slide 5
Example 1: Which table of values represents a linear relation?
Justify the answer.
Celsius
(C)
Fahrenheit
(F)
0 31
5 41
10 50
15 59
20 68
Amps
(I)
Watts
(P)
0 0
5 75
10 300
15 675
20 1200
+5
+5
+5
+5
+9
+9
+9
+9
This is a linear relation.
The changes in both
variables are constant.
+5
+5
+5
+5
+75
+225
+375
+525
This is NOT a linear relation.
The changes in P are not
constant.

6. Slide 6
Example 1: Graph the equation and state if it represents a linear
relation.
3 25y x  
x y
-3
-1
0
1
3

7. Slide 7
Example 1: Graph the equation and state if it represents a linear
relation.
3 25y x  
x y
-3 34
-1 28
0 25
1 22
3 16
Answer:
The above equation is a linear relation.

8. Slide 8
Example 2: Graph the equation and state if it represents a linear
relation.
2
2 5y x 
x y
-3
-1
0
1
3

9. Slide 9
Example 2: Graph the equation and state if it represents a linear
relation.
2
2 5y x 
x y
-3 23
-1 7
0 5
1 7
3 23
Answer:
The above equation is NOT a linear relation (not a straight line).

10. Slide 10
Example 3: Graph the equation and state if it represents a linear
relation.
5y 
x y
-3
-1
0
1
3

11. Slide 11
Example 3: Graph the equation and state if it represents a linear
relation.
5y 
x y
-3 5
-1 5
0 5
1 5
3 5
Answer:
The above equation is a linear relation.

12. Slide 12
Example 3: Graph the equation and state if it represents a linear
relation.
4x  
x y
-3
-1
0
1
3

13. Slide 13
Example 3: Graph the equation and state if it represents a linear
relation.
4x  
x y
-4 -3
-4 -1
-4 0
-4 1
-4 3
Answer:
The above equation is a linear relation.

14. Change of Rate
Slide 14
𝑅𝑎𝑡𝑒 𝑜𝑓 𝐶ℎ𝑎𝑛𝑔𝑒 =
𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝐷𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒
𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝐼𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒

15. Slide 15
Example 1: This graph shows the altitude of
a helicopter as it lifts off.
a) Identify the dependent and independent
variables.
b) Find the rate of change. What does it
represent?

16. Slide 16
Example 1: This graph shows the altitude of
a helicopter as it lifts off.
a) Identify the dependent and independent
variables.
b) Find the rate of change. What does it
represent?
The dependent variable is the Altitude.
The independent variable is Time.

17. Slide 17
Example 1: This graph shows the altitude of
a helicopter as it lifts off.
a) Identify the dependent and independent
variables.
b) Find the rate of change. What does it
represent?
The dependent variable is the Altitude.
The independent variable is Time.
𝑅𝑎𝑡𝑒 𝑜𝑓 𝐶ℎ𝑎𝑛𝑔𝑒 =
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑑𝑒𝑝𝑒𝑛𝑑. 𝑣𝑎𝑟.
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑. 𝑣𝑎𝑟.
𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑑𝑒𝑝𝑒𝑛𝑑. 𝑣𝑎𝑟. = 240 𝑚 − 120 𝑚 = 120 𝑚
𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑. 𝑣𝑎𝑟. = 120 𝑠 − 60 𝑠 = 60 𝑠
𝑅𝑎𝑡𝑒 𝑜𝑓 𝐶ℎ𝑎𝑛𝑔𝑒 =
120 𝑚
60 𝑠
= 2 𝑚/𝑠 Answer:
The rate of change is 2 m/s. It
represents the helicopter
climbing 2 metres every second.

18. Slide 18
Example 2: This graph shows the altitude of
a helicopter as it descents to land.
a) Identify the dependent and independent
variables.
b) Find the rate of change. What does it
represent?

19. Slide 19
Example 2: This graph shows the altitude of
a helicopter as it descents to land.
a) Identify the dependent and independent
variables.
b) Find the rate of change. What does it
represent?
The dependent variable is the Altitude.
The independent variable is Time.
𝑅𝑎𝑡𝑒 𝑜𝑓 𝐶ℎ𝑎𝑛𝑔𝑒 =
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑑𝑒𝑝𝑒𝑛𝑑. 𝑣𝑎𝑟.
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑. 𝑣𝑎𝑟.
𝑅𝑎𝑡𝑒 𝑜𝑓 𝐶ℎ𝑎𝑛𝑔𝑒 =
180 𝑚 − 0 𝑚
60 𝑠 − 120 𝑠
=
180 𝑚
−60 𝑠
= −3 𝑚/𝑠
Answer:
The rate of change is -3 m/s. It represents the helicopter
descending 3 metres every second.