Question

Determine the rate of change and the initial value of the linear function given here as a table

Answer 1

To determine the rate of change and the initial value of the linear function from a table, we need to calculate the difference in the y-values (dependent variable) between two points and then divide that by the difference in the x-values (independent variable).

Let's consider the following table:

| x | y |
|---|---|
| 1 | 3 |
| 2 | 6 |
| 3 | 9 |

To calculate the rate of change, we will use the formula:

Rate of change = (change in y) / (change in x)

For points (1, 3) and (2, 6):
Rate of change = (6 - 3) / (2 - 1) = 3 / 1 = 3

Now, for points (2, 6) and (3, 9):
Rate of change = (9 - 6) / (3 - 2) = 3 / 1 = 3

Since the rate of change is constant across all points, we can conclude that the rate of change is 3.

To find the initial value of the linear function, we can substitute the rate of change and any point from the table into the slope-intercept form of a linear equation: y = mx + b, where m is the rate of change and b is the initial value.

Let's use the point (1, 3) to find the initial value:

3 = 3(1) + b
b = 3 - 3
b = 0

Therefore, the initial value of the linear function is 0.

In conclusion, the rate of change of the linear function is 3, and the initial value is 0.