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)
(x_{1}, y_{1}) ==> (0, 0) and (x_{2}, y_{2}) ==> (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) = (y_{2} - y_{1}) / (x_{2} - x_{1})
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) = (y_{2} - y_{1}) / (x_{2} - x_{1})
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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