The table represents a linear function. The rate of change between the points $(-5,10)$ and $(-4,5)$ is -5. What is the rate of change between the points $(-3,0)$ and $(-2,-5)$?
-5
-\frac{1}{5}
\frac{1}{5}
5
Answer (1)
The rate of change between two points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by x 2 − x 1 y 2 − y 1 .
Substitute the given points ( − 3 , 0 ) and ( − 2 , − 5 ) into the formula.
Calculate the rate of change: − 2 − ( − 3 ) − 5 − 0 = 1 − 5 = − 5 .
The rate of change between the points ( − 3 , 0 ) and ( − 2 , − 5 ) is − 5 .
Explanation
Understanding the Problem We are given a table representing a linear function and asked to find the rate of change between the points ( − 3 , 0 ) and ( − 2 , − 5 ) . The rate of change, also known as the slope, measures how much the y -value changes for each unit change in the x -value.
Applying the Formula The rate of change between two points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by the formula: Rate of change = x 2 − x 1 y 2 − y 1 In our case, ( x 1 , y 1 ) = ( − 3 , 0 ) and ( x 2 , y 2 ) = ( − 2 , − 5 ) .
Calculating the Rate of Change Plugging in the values, we get: Rate of change = − 2 − ( − 3 ) − 5 − 0 = − 2 + 3 − 5 = 1 − 5 = − 5 So, the rate of change between the points ( − 3 , 0 ) and ( − 2 , − 5 ) is − 5 .
Final Answer The rate of change between the points ( − 3 , 0 ) and ( − 2 , − 5 ) is − 5 .
Examples
Understanding rate of change is crucial in many real-world applications. For example, if you're tracking the distance a car travels over time, the rate of change represents the car's speed. If the car travels 100 miles in 2 hours, the rate of change (speed) is 50 miles per hour. Similarly, in business, the rate of change can represent the growth rate of revenue or the depreciation rate of an asset. Knowing how to calculate and interpret rate of change helps in making informed decisions and predictions.
