Numbers, Relations & Functions 10 Section 5.6 Properties of Linear Relations
We can identify a linear relation in different ways.
a table of values • a graph
Number Profit ($) sold 0 –800 200 0 400 800 600 1600 a set of ordered pairs
{(0, –800), (200, 0), (400, 800), (600, 1600)}
For a linear relation, a constant change in the independent variable results in a constant change in the dependent variable.
We can use each representation above to calculate the rate of change (how one quantity changes in relation to another quantity):
change in dependent variable Rate of Change = change in idependent variable
In the above example,
change in dependent variable Rate of Change = change in idependent variable
=
=
The equation for this linear function, where P represents profit in dollars and n represents the number of baseball caps sold, is:
P = 4n – 800 Numbers, Relations & Functions 10 Section 5.6
Example 1: Determining whether a Table of Values Represents a Linear Relation
Which table of values represents a linear relation? Justify your answer.
The relation between the amount of tax The relation between the number of bacteria charged, T dollars, and the amount of in a culture, n, and time, t minutes. the purchase, A dollars.
The terms in the first columns are in numerical order. So, calculate the change in each variable.
Since the changes in both variables are The changes in t are constant, but the changes ______, the table of values in n are ______, so the table of ______a linear relation values ______represent a linear relation.
Example 2: Determining whether an Equation Represents a Linear Relation
Set up a table of values for and graph each relation. Which equations represent linear relations? How do you know?
a. y= -2 x + 1
x y Numbers, Relations & Functions 10 Section 5.6
b. y= -3 x2 + 9
x y
c. y = 4 d. x = -3
x y Numbers, Relations & Functions 10 Section 5.6
The graphs for a, c, and d are ______, so their equations represent linear relations; that is ______, ______, and ______.
Example 3: Identifying a Linear Relation
Which of the following relations is linear? Justify your answer.
a. A classic car originally purchased for $25 000 appreciates in value by 10% every year. The car’s value is related to time.
b. Laura charges $20 per hour for tutoring. Her fee is related to time.
Solution:
Create a table of values, then check to see if the relation is linear.
a. Every year, the car’s value increases by 10%, so the value is 100% + 10% = 110% of its value the previous year. So, multiply the car’s value the previous year by ______.
Time (years) Value ($) 0 25 000 1 2 3
There is a constant change of _____ in the first column, but the differences in the second
column are not constant, so the relation is ______.
b. Every hour, Laura’s total fee increases by $20.
Time (hours) Fee ($) 0 1 2 3
There is a constant change of _____ in the first column and a constant change of _____ in
the second column, so the relation is ______. Numbers, Relations & Functions 10 Section 5.6
Example 4: Determining the Rate of Change of a Linear Relation From its Graph
A hot tub contains 1600 L of water. Graph A represents the hot tub being filled at a constant rate. Graph B represents the hot tub being emptied at a constant rate.
a. Identify the dependent and independent variables. b. Determine the rate of change of each relation and explain what it represents.
Solution:
a. The independent variable is ______. The dependent variable is ______.
b. Choose two points on the line. Calculate the change in each variable from one point to the other.
Graph A
change in dependent variable Rate of Change = change in idependent variable
=
=
The rate of change is positive, so the volume is ______with time at a rate of ______.
Graph B Numbers, Relations & Functions 10 Section 5.6
change in dependent variable Rate of Change = change in idependent variable
=
=
The rate of change is negative, so the volume is ______with time at a rate of ______.