For linear functions, the rate of change, denoted by m, is represented in the slopeintercept form of a line: y = mx + b. The rate of change of functions is generally defined as (f(b)  f(a)) / (b  a), where 'a' and 'b' are two input values of the independent variable, and f(a) and f(b) are their corresponding output values of the dependent variable.
Rate of Change Formula
To calculate the rate of change, we need to find the change in distance divided by the change in time.
Change in distance = 180  40 = 140 miles
Change in time = 4  2 = 2 hours
Therefore, the rate of change of distance with respect to time is:
Rate of change = Change in distance / Change in time
Rate of change = 140 miles / 2 hours
Rate of change = 70 miles per hour
Hence, the rate of change for the given information is 70 miles per hour.
Example 2: Calculate the rate of change for the following information in the table:
To calculate the rate of change, we need to find the change in height divided by the change in time.
Change in height = 7  4 = 3 inches
Change in time = 140  50 = 90 days
Therefore, the rate of change of the height of the tree with respect to time is:
Rate of change = Change in height / Change in time
Rate of change = 3 inches / 90 days
Rate of change = 0.0333 inches per day (rounded to four decimal places)
Hence, the rate of change for the given information is 0.0333 inches per day.
Example 3: Find the rate of change for the situation: Ron completed 3 math assignments in one hour and Duke completed 6 assignments in two hours.
To find the rate of change, we need to determine the change in the number of math assignments completed by each person per hour.
For Ron:
Number of assignments completed = 3
Time taken = 1 hour
Therefore, the rate of change for Ron can be calculated as:
Rate of change = Number of assignments completed / Time taken
Rate of change = 3 / 1
Rate of change = 3 assignments per hour
For Duke:
Number of assignments completed = 6
Time taken = 2 hours
Therefore, the rate of change for Duke can be calculated as:
Rate of change = Number of assignments completed / Time taken
Rate of change = 6 / 2
Rate of change = 3 assignments per hour
Hence, the rate of change for both Ron and Duke is 3 assignments per hour.
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