Find the rate of change represented by the data in the table below.
\begin{tabular}{c|ccccc}
$x$ & 0 & 3 & 6 & 9 & 12 \\
\hline
$y$ & 9 & 8 & 7 & 6 & 5
\end{tabular}
Is the rate of change positive or negative?
a) positive
b) negative
Answer (1)
Calculate the rate of change using two points from the table: 3 − 0 8 − 9 = − 3 1 .
Determine that the rate of change is negative since − 3 1 < 0 .
The rate of change is − 3 1 .
The rate of change is negative. − 3 1 and 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 .
Calculating the Rate of Change To calculate the rate of change, we can use any two points from the table. Let's use the points ( 0 , 9 ) and ( 3 , 8 ) . The rate of change is given by the formula: Rate of Change = Change in x Change in y = x 2 − x 1 y 2 − y 1 Substituting the values, we get: Rate of Change = 3 − 0 8 − 9 = 3 − 1 = − 3 1 So, the rate of change is − 3 1 .
Determining the Sign of the Rate of Change Since the rate of change is − 3 1 , which is a negative value, the rate of change is negative.
Final Answer The rate of change is − 3 1 , and it is negative. Therefore, the correct answer is: Rate of Change: − 3 1 Sign: negative
Examples
Understanding rate of change is crucial in many real-world applications. For instance, consider a car's fuel efficiency. If a car travels 300 miles on 10 gallons of gas, the rate of change (miles per gallon) is 30. This concept helps in predicting how far the car can travel on a given amount of fuel. Similarly, in business, understanding the rate of change in sales or profits helps in making informed decisions about investments and strategies. The rate of change is a fundamental concept in calculus and is used extensively in physics, engineering, economics, and many other fields.
