Linear Relations
Direct Variation versus Partial Variation
Linear Relations Day 1
Name: _______________________
MPM1D
Date _________________________
Linear Relation – A relation between two variables that appears as a straight line when graphed.
Example 1: Lucy can jog at a steady pace of 150 m/min.
a) Create a table of values.
Time (min)
Distance (m)
0
1
2
3
4
5
b) Graph.
c) Where does the graph intersect the y-axis? __________ Called the initial value.
d) Write an equation to determine the distance, d, in metres that Susan jogs in t minutes.
_________________________________ The relation is a ___________ variation.
e) Consider the distance Lucy jogged in 2 min.
What happens to the distance when the time is doubled? __________________
Example 2: Frank works part-time at a local donut shop. He earns $10/h.
a) Create a table of values.
Time Worked, t
(hours)
Pay, P
($)
0
2
4
6
8
10
b) Graph.
c) What is the initial value? _________
d) Write an equation to determine his pay, P, in dollars, and the time, t, in hours, he works.
___________________________ The relation is a ___________ variation.
e) One week, Frank works 9 h. Find his pay for that week. (Two ways ~ graph or equation)
Example 3: The distance traveled by a train varies directly with time. The train travels 180 km in 2 h.
Determine the constant of variation and write an equation for the relationship.
Example 4: A taxi ride costs $3 plus $2 per kilometre.
a) Create a table of values.
Distance, d
(km)
Cost, C
($)
0
1
2
3
4
5
b) Graph.
c) What is the initial value? _____
d) What is the rate of change? __________________
e) Write an equation for the relationship.
____________________ This relation is a ______________ variation.
Example 5:
x
y
0
6
b) Identify the initial value. ________
1
9
c) Identify the rate of change. __________
2
a) Complete the table of values given that y varies partially with x.
d) Write an equation relating y and x in the form y = mx + b.
____________________
3
15
4
21
SUMMARY:
DIRECT VARIATION
! One variable is a constant multiple of the other.
! The initial value is zero. It passes through the origin.
! Constant of variation, k, is the constant multiple.
! Equation is y=kx
PARTIAL VARIATION
One variable varies by a multiple of the other variable PLUS a constant.
It has an initial value, b. It passes through the y-axis, NOT the origin.
Rate of change, m, is also called the constant of variation.
Equation is y=mx+b where m is the constant of variation and b is the initial value (fixed value)
HOMEWORK: Workbook
Page 81 Questions 1-4 AND page 83 Questions 1-3