COMPARING LINEAR FUNCTIONS

A rate of change is a rate that describes how one quantity changes in relation to another quantity. If x is the independent variable and y is the dependent variable, then

Rate of change = Change in y/ Change in x

Examine each set of functions and determine which has the greater rate of change, if either. Explain your reasoning.

Problem 1 :

Solution :

From function A :

y = 8x - 3

Comparing the given equation with slope intercept form

y = mx + b

m = 8

From function B :

Taking two points on the graph, (0, 0) and (2, 2)

(x1, y1) ==> (0, 0) and (x2, y2) ==> (2, 2)

Change in y = (2 - 0) ==> 2

Change in x = (2 - 0) ==> 2

Slope = Change in y / change in x ==> 2/2 ==> 1

Function A is having greater rate of change.

Problem 2 :

Solution :

Rate of change from function C :

Taking two points from the table, (-1, -6) and (0, -3).

Slope (m) = (y2 - y1) / (x2 - x1)

m = (3 + 6) / (0 + 1)

m = 9/1

m = 9

Rate of change from function D :

Taking two points from the table, (0, 3) and (6, 0).

Slope (m) = (y2 - y1) / (x2 - x1)

m = (0 - 3) / (6 - 0)

m = -3/6

m = -1/2

Function C is having greater rate of change.

Problem 3 :

Solution : 

Rate of change from equation E :

6x + y = 1

Converting the given equation into slope intercept form, we get

y = -6x + 1

Slope (m) = -6

Rate of change from equation F :

Two points from the graph (-1, -6) and (1, 6).

Change in y = (6 +6) ==> 12

Change in x = (1 + 1)  ==>  2

Slope (m) = Change in y / change in x

Slope (m) = 12/2

m = 6

Equation F is having grater rate of change.

Problem 4 :

Alicia, Cherie, and John had been studying rates of change and were discussing the best way to determine which linear function has the greater rate of change.

Solution :

Rate of change from G :

Considering two points on the line (0, 1) and (4, 3).

Change in y = (3 - 1) ==> 2

Change in x = (4 - 0) ==> 4

Slope = 2/4 ==> 1/2

Rate of change from H :

Two points from the table are (0, 1) and (1, 3).

m = (3 - 1) / (1 - 0)

m = 2/1

m = 2

So, function H is having greater rate of change.

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