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Day 7 – Characteristics of Linear Relations Minds On A car rental company charges $50 per day to rent a mid sized car plus an administrative fee of $40 for reserving the car. This statement is a representation of the relation using words. Represent the relation using a table, a graph and an equation. Table: Graph: Days (d) 0 1 2 3 4 5 Cost (C) 300 250 200 150 100 50 2 Equation: 4 6 Part A: Write an equation when a relation is represented in words The equation for a linear relation: y mx b Considering that a line is really just a set of ordered pairs, (x , y), it makes sense that the equation of a line needs to contain the variables x and y. These variables will define the coordinates that make up the line. This means that the only 2 values that need to be determined in order to write the equation of a linear relation are m and b. When a linear relation is represented in words m is the rate of change and b is the initial value. Linear relation represented in words: m = rate of change b = initial value Example 1: Write an equation for the following relationship by first identifying the value of m and b. The Copy Centre charges $75 to design a poster and 25 cents for each copy. m= b= And the equation of this linear relation is: If The Copy Centre changed their cost per flyer to 35 cents for each copy the equation would become: If The Copy Centre changed their design cost to $125 the equation will become: Part B: Write an equation when a relation is represented in a table of values The equation for a linear relation: y mx b m The rate of change can be determined using a table with: y x The initial value can be found in the table when the independent variable ”x” is zero. b = the value of y when x is set to zero Example 2: a) Determine the equation of the following linear relations using the tables provided: x 0 1 2 3 4 y 6 9 12 15 18 b) x -2 0 2 4 6 m= m= b= b= Equation: Equation: y 3 5 7 9 11 What should we do if the initial value isn’t in the table? Example 3: c) Determine the equation of the following linear relations using the tables provided: x y 2 3 4 -11 -8 -5 d) x -8 -6 -4 m= m= b= b= Equation: Equation: y -5 -10 -15 Part C: Write an equation when a relation is represented as a graph y mx b The equation for a linear relation: The rate of change can be determined on a graph with: m rise run The initial value can be found on the graph by finding the value of the dependent variable “y” when x is zero, this value occurs when: b = value of y when the line crosses the y-axis (y-intercept) Example 4: Determine the equation of each of the following linear relations. a) 6 b) 6 y 4 4 2 2 y x -5 -5 5 -2 -2 -4 -4 m= m= b= b= Equation: Consolidate: x 5 Equation: Complete the following table to show how to find the equation of a line using different representations. When represented in words When represented in a table When represented as a graph To find m To find b Replace the m and b with their values in the general equation of a linear relation: y mx b