Find the rate of change represented by the data in the table below.
\begin{tabular}{l|lllll}
$x$ & -8 & -4 & 0 & 4 & 8 \\
\hline
$y$ & -5 & -2 & 1 & 4 & 7
\end{tabular}
Is the rate of change positive or negative?
a) positive
b) negative
Rate of Change: $\frac{\text { change in } y}{\text { change in } x}$
Answer (1)
Calculate the rate of change using two points from the table: x 2 − x 1 y 2 − y 1 .
Substitute the values from points ( − 8 , − 5 ) and ( − 4 , − 2 ) : − 4 − ( − 8 ) − 2 − ( − 5 ) = 4 3 .
Verify the rate of change with points ( 0 , 1 ) and ( 4 , 4 ) : 4 − 0 4 − 1 = 4 3 .
Determine that the rate of change is positive since 0"> 4 3 > 0 , and the rate of change is 4 3 .
Explanation
Understanding the Problem We are given a table of x and y values and asked to find the rate of change represented by the data. The rate of change is defined as the change in y divided by the change in x , which can be calculated using the formula: change in x change in y = x 2 − x 1 y 2 − y 1 We also need to determine if the rate of change is positive or negative.
Calculating the Rate of Change To find the rate of change, we can choose any two points from the table. Let's choose the first two points: ( − 8 , − 5 ) and ( − 4 , − 2 ) . Plugging these values into the formula, we get: − 4 − ( − 8 ) − 2 − ( − 5 ) = − 4 + 8 − 2 + 5 = 4 3 = 0.75
Verifying the Rate of Change We can verify this by choosing another pair of points, say ( 0 , 1 ) and ( 4 , 4 ) . The rate of change is: 4 − 0 4 − 1 = 4 3 = 0.75 The rate of change is constant, which means the relationship is linear.
Determining the Sign of the Rate of Change Since the rate of change is 4 3 , which is a positive number, the rate of change is positive.
Final Answer The rate of change is 4 3 , and it is positive. Therefore, the answer to the first part of the question is 4 3 , and the answer to the second part is (a) positive.
Examples
Imagine you're tracking the distance you travel over time. The rate of change, in this case, represents your speed. If you travel 3 miles every 4 hours, your speed (rate of change) is 3/4 miles per hour. Understanding rate of change helps in many real-life situations, such as calculating speed, determining the slope of a hill, or analyzing the growth of a plant over time. This concept is fundamental in physics, engineering, and economics.
