Question

Which functions have a positive rate of change and which have a negative rate of change?

A.) 2x – 3y = 9

B.) y = −4x + 5

C.)
x y
−1 −5
1 1
3 7

D.)
x y
1 −2
2 −6
3 −10

Answer 1

The rate of change of a plot or equation can be determined by determining its slope. a positive rate of change pertains to a positive slope, otherwise a negative slope.

1. 2x – 3y = 9 ; m = 2/3 -- positive rate of change
2. y = −4x + 5 ; m =-4; --- negative rate of change
3. y = -2 + 3 x ; m = 3 ---- positive rate of change
4. y = 2 -4x ; m =-4; --- negative rate of change

Answer 2

Rate of change: (f(2) - f(1))/(2-1)
Function A: 2x - 3y = 9
3y = 2x - 9
y = 2/3 x - 3
f(2) = 2/3 (2) - 3 = 4/3 - 3 = -5/3
f(1) = 2/3 (1) - 3 = 2/3 - 3 = -7/3
Rate of change = -5/3 - (-7/3) = -5/3 + 7/3 = 2/3 (positive)

Function B: y = -4x + 5
f(2) = -4(2) + 5 = -8 + 5 = -3
f(1) = -4(1) + 5 = -4 + 5 = 1
Rate of change = -3 - 1 = -4 (negative)

Function C: Rate of change = (f(1) - f(-1))/(1-(-1)) = (1 - (-5))/(1 + 1) = (1 + 5)/2 = 6/2 = 3 (positive)

Function D: Rate of change = -6 - (-2) = -6 + 2 = -4 (negative)

Therefore, functions A and C have positive rates of change while functions B and D have negative rates of change.