Investigating Properties of
Linear Relations
Cole buying a car.
Cole bought a new car for $27,500. This
graph shows it’s value over the first three
years. When will Cole’s car be worth $0.
Value of Car by Age
Value of car ($)
30000
25000
20000
15000
10000
5000
0
0
1
2
3
4
Age of car (years)
5
6
Calculate the amount by which Cole’s car
decreased in value between years 1 and 2.
Value Year 2 – Value Year 1 = 17,500 – 22,500 = -5000
The value of the car decreased by approximately $5000.
Calculate the rate of change in the car’s
value between years 1 and 3.
Value Year 3 – Value Year 1 = 12,500 – 22,500 = -10000
The value of the car decreased by approximately $10000.
Slope (Note)
• A measure, often represented by m, of the
steepness of a line
• The ratio comparing the vertical and horizontal
distances (called the rise and run)
rise y change in y
m


run x change in x
Calculate the slope of the graph between
years 1 and 3.
y 22,500  12,500 10,000
m


 5000
x
1 3
2
This means that the value of the car is
decreasing by $5000 a year (approximately).
How does the slope compare to your answer in
part 1?
It is the same!
Complete the first difference column in the
table below. How does the first differences
compare to the slope in part 3?
Age of Car Value of Car
Year (x)
$ (y)
0
1
27,500
First Difference
$ (Δy)
22,675 – 27,500 = -4825
22,675
17,850 – 22,675 = -4825
2
17,850
3
13,025
13,025 – 17,850 = -4825
8,200 – 13,025 = -4825
4
8,200
They are
approximately
the same! A
little different
because we
approximated
from the
graph.
Complete the following table. Why are the
first differences different than in part 4?
Age of
Car
Year (x)
Value of Car
$ (y)
0
27,500
2
17,850
First Difference
$ (Δy)
17,850 – 27,500 = -9650
8,200 – 17,850 = -9650
4
8,200
-1,450 – 8,200 = -9650
6
-1450
Because the
values of x
(Age of the
Car) are
increasing by
2 instead of
by 1.
Write an equation for the relation between
the car’s value and it’s age.
y=27,500 – 4,825x
Y=-4,825x + 27,500
Slope
First Differences
Y-Intercept
Determine the x-intercept of the graph. Use
it to tell when Cole’s car will be worth $0.
How do you know?
The x-intercept
appears to be
approximate 5.75
years.
It is where the
graph crosses
the x-axis.
What is the connection between
the first differences and the slope?
• The first differences equal the slope!
• BUT – only the difference in the x values are 1!!
When you calculated the slope, did it matter
which points you chose? Explain.
No, it does not!
With a linear relationship, the
slope is constant.
Use the graph to explain why the
first differences were constant.
Value of Car by Age
Value of car ($)
30000
25000
20000
15000
10000
5000
0
0
1
2
3
4
5
6
Age of car (years)
A linear relationship has a constant slope … and
therefore, constant first differences.