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
Rate of change from two points (x_{1}, y_{1}) and (x_{2}, y_{2}) :
Using the formula given below, we can find rate of change.
m = (y_{2} - y_{1})/(x_{2} - x_{1})
(x_{1}, y_{1}) ==> (-1, 15) and (x_{1}, y_{1}) ==> (0, 25)
m = (25 - 15) / (0 + 1)
m = 10/1
m = 10
Rate of change is positive.
Rate of change from graph :
To find slope or rate of change from the graph of the line, we can take two points on the line and horizontal change and vertical change.
Rate of change = Rise / Run
Take two points from the line and calculate horizontal and vertical changes.
m = -3/6
m = -1/2
Rate of change from the equation :
If the equation is in the form y = mx + b
m is known as slope and b is y-intercept.
Examine each set of functions and determine which has the greater rate of change, if either. Explain your reasoning.
Problem 1 :
Solution :
Rate of change from Table A :
Take any two points from table A. Say (-2, -8) and (2, -5).
(x_{1}, y_{1}) ==> (-2, -8) and (x_{1}, y_{1}) ==> (2, -5)
m = (y_{2} - y_{1})/(x_{2} - x_{1})
m = (-5 + 8) / (2 + 2)
m = 3/4
Rate of change from Table B :
Take any two points from table A. Say (-5, -46) and (1, -38).
(x_{1}, y_{1}) ==> (-5, -46) and (x_{1}, y_{1}) ==> (1, -38)
m = (y_{2} - y_{1})/(x_{2} - x_{1})
m = (-38+46) / (1+5)
m = 8/6
m = 3/4
Both are having the same rate of change.
Problem 2 :
Solution :
Taking two points from Graph A :
(0, 3) and (6, 0)
Change in y = Rise = 0 - 3 ==> -3
Change in x = Run = 6 - 0 ==> 6
Slope = Rise / Run ==> -3/6 ==> -1/2
Taking two points from Graph B :
(0, 7) and (2, 3)
Change in y = Rise = 3 - 7 ==> -4
Change in x = Run = 2 - 0 ==> 2
Slope = Rise / Run ==> -4/2 ==> -2
Graph A is having greater rate of change.
Problem 3 :
Equation A : 5x + 6y = 60
Equation B : y = (-1/4)x - 2
Solution :
From equation A :
To find rate of change from Equation A , we can convert the given equation into slope intercept form.
6y = -5x + 60
Divide by 6 on both sides.
y = (-5/6)x + (60/6)
y = (-5/6)x + 10
Rate of change = -5/6
From equation B :
y = (-1/4)x - 2
Rate of change = -1/4
Rate of change in equation B is greater.
Problem 4 :
An ice cream shop is choosing a milk delivery service. The Spotted Cow charges $2.80 per gallon, plus a $2 delivery fee. Dairy Farms charges $2.10 per gallon, plus a $10 delivery fee.
Solution :
Let x be the charges per gallon and y be the total charge.
Creating the equation for the given situation, we get
Spotted cow charges :
y = 2.80x + 2
Rate of change = 2.80
Dairy Farms charges :
y = 2.10x + 10
Rate of change = 2.10
Spotted cow charge is greater.
May 21, 24 08:51 PM
May 21, 24 08:51 AM
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