Find the rate of change represented by the data in the table below.
\begin{tabular}{c|ccccc}
x & -14 & -7 & 0 & 7 & 14 \
\hline y & 1 & -1 & -3 & -5 & -7
\end{tabular}
Is the rate of change positive or negative?
a) positive
b) negative
Answer (2)
Calculate the rate of change using two points from the table: Δ x Δ y = − 7 − ( − 14 ) − 1 − 1 = 7 − 2 .
Verify the rate of change is constant using other pairs of points.
Determine that the rate of change is negative since − 7 2 < 0 .
State the rate of change and indicate that it is negative: − 7 2 , negative.
Explanation
Understanding the Problem We are given a table of x and y values and asked to find the rate of change and determine if it is positive or negative. The rate of change is calculated as the change in y divided by the change in x , which can be written as Δ x Δ y .
Calculating Rate of Change Let's calculate the rate of change using the first two points in the table, ( − 14 , 1 ) and ( − 7 , − 1 ) . The change in y is − 1 − 1 = − 2 , and the change in x is − 7 − ( − 14 ) = − 7 + 14 = 7 . Therefore, the rate of change is 7 − 2 = − 7 2 .
Verifying Constant Rate of Change To verify that the rate of change is constant, let's calculate it using the next two points, ( − 7 , − 1 ) and ( 0 , − 3 ) . The change in y is − 3 − ( − 1 ) = − 3 + 1 = − 2 , and the change in x is 0 − ( − 7 ) = 0 + 7 = 7 . Therefore, the rate of change is 7 − 2 = − 7 2 .
Further Verification We can also calculate the rate of change using the points ( 0 , − 3 ) and ( 7 , − 5 ) . The change in y is − 5 − ( − 3 ) = − 5 + 3 = − 2 , and the change in x is 7 − 0 = 7 . Therefore, the rate of change is 7 − 2 = − 7 2 . Finally, using the points ( 7 , − 5 ) and ( 14 , − 7 ) . The change in y is − 7 − ( − 5 ) = − 7 + 5 = − 2 , and the change in x is 14 − 7 = 7 . Therefore, the rate of change is 7 − 2 = − 7 2 .
Determining the Sign Since the rate of change is consistently − 7 2 , it is negative. Therefore, the answer to the first part of the question is negative.
Final Answer The rate of change is − 7 2 and it is negative.
Examples
Understanding rate of change is crucial in many real-world scenarios. For instance, consider a car traveling at a constant speed. The rate of change (speed) can be calculated by dividing the distance traveled (change in position) by the time taken (change in time). If a car travels 100 miles in 2 hours, the rate of change is 50 miles per hour. Similarly, in economics, the rate of change can represent the inflation rate, calculated as the percentage change in the price index over a period. These concepts help us analyze and predict changes in various situations.
The rate of change calculated from the data table is − 7 2 , which is negative. This was confirmed by using multiple pairs of points from the table, where each yielded the same result. Therefore, the rate of change is negative.
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