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Recognize constant rate of change

A constant rate of change means that the dependent quantity changes by the same amount for each equal change in the independent quantity; equivalently, the ratio of change in output to change in input remains constant across intervals. It can be identified in tables, graphs as a straight line, and equations of the form y=mx+by=mx+b, where mm represents the rate and may be positive, negative, or zero; the relationship need not be proportional or pass through the origin.

Detailed Explanation: Recognize constant rate of change

A relationship has a constant rate of change when the output changes by the same amount whenever the input changes by the same amount.

To check a table:

  1. Find the change in the input values.
  2. Find the change in the output values.
  3. Divide the output change by the input change:
rate of change=change in ychange in x\text{rate of change}=\frac{\text{change in }y}{\text{change in }x}
  1. If the rate is the same for every interval, the relationship has a constant rate of change.

Example

xxyy
0055
1188
221111
331414

The input increases by 11 each time. Now compare the output changes:

  • From 55 to 88: 8−5=38-5=3
  • From 88 to 1111: 11−8=311-8=3
  • From 1111 to 1414: 14−11=314-11=3

The output increases by 33 for every increase of 11 in the input. The rate of change is

31=3.\frac{3}{1}=3.

Therefore, this relationship has a constant rate of change of 33.

It can be written as

y=3x+5.y=3x+5.

The 33 shows the constant rate of change. The relationship does not need to pass through (0,0)(0,0); here, when x=0x=0, y=5y=5.

Learn by doing: Recognize constant rate of change

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