Question

This table shows input and output values for a linear function f(x) .

x -5 -4 -3 -2 -1 0 1 2 3
f(x) -11 -3 5 13 21 29 37 45 53

What is the positive difference of outputs for any two inputs that are four values apart?

Answer 1

The positive difference of outputs for any two inputs that are four units apart in a linear function is constant, and it can be found by subtracting the outputs of these inputs. For instance, with inputs x = -4 and x = 0, their respective outputs are -3 and 29, the positive difference is 32. This example demonstrates the consistent rate of change characteristic of linear functions.

Step-by-step explanation:

The student is asking about the positive difference of outputs for a linear function when two input values are four units apart. In the given table, we can pick any two x-values that are four units apart and then look at the corresponding f(x) values to calculate the positive difference in outputs. To illustrate with an example, if we take x = -4 and x = 0 (which are four units apart), their respective outputs from the table are f(-4) = -3 and f(0) = 29. The positive difference between these outputs is 29 - (-3) = 32.

We can perform the same calculation for any other pair of inputs that are four units apart, and because the function is linear, the positive difference will always be the same. This is due to the constant rate of change in a linear function, which is also known as the slope of the function. If a linear function has the table's pattern of a steady increase or decrease between consecutive inputs, the difference in outputs for inputs four units apart will always be the same.