COMPARING RATE OF CHANGE

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 (x₁, y₁) and (x₂, y₂) :

Using the formula given below, we can find rate of change.

m = (y₂ - y₁)/(x₂ - x₁)

(x₁, y₁) ==> (-1, 15) and (x₁, y₁) ==> (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

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).

(x₁, y₁) ==>  (-2, -8)  and (x₁, y₁) ==> (2, -5)

m = (y₂ - y₁)/(x₂ - x₁)

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).

(x₁, y₁) ==>  (-5, -46) and (x₁, y₁) ==> (1, -38)

m = (y₂ - y₁)/(x₂ - x₁)

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.