Question

find rate of change from a table

1. find sophia’s average rate of change in earnings per hour.

sophia’s earnings
hours (x) | earnings (y)
5 | $126
10 | $252
15 | $378
20 | $504
roc: _____________

1. find parker’s average rate of change in miles per gallon.

parker’s gas mileage
gallons (x) | miles (y)
20 | 722
40 | 1444
60 | 2166
80 | 2888
roc: _____________

Explanation:

Step1: Recall Rate of Change Formula

The average rate of change between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(\frac{y_2 - y_1}{x_2 - x_1}\). For a linear relationship (constant rate of change), we can use any two points.

Step2: Calculate Sophia's Rate of Change

Take two points from Sophia's Earnings table, e.g., \((5, 126)\) and \((10, 252)\).

$$ \text{ROC} = \frac{252 - 126}{10 - 5} = \frac{126}{5} = 25.2 $$

We can verify with another pair, say \((10, 252)\) and \((15, 378)\):

$$ \frac{378 - 252}{15 - 10} = \frac{126}{5} = 25.2 $$

Step3: Calculate Parker's Rate of Change

Take two points from Parker's Gas Mileage table, e.g., \((20, 722)\) and \((40, 1444)\).

$$ \text{ROC} = \frac{1444 - 722}{40 - 20} = \frac{722}{20} = 36.1 $$

Verify with \((40, 1444)\) and \((60, 2166)\):

$$ \frac{2166 - 1444}{60 - 40} = \frac{722}{20} = 36.1 $$

Answer:

Sophia's average rate of change: \(\$25.2\) per hour
Parker's average rate of change: \(36.1\) miles per gallon