Question

Linear function A and linear function B both have the same input values as shown below. Why will the output values for linear function A always be different than the corresponding output values for linear function B?

2 tables. A 2-column table with 5 rows titled Linear Function A. Column 1 is labeled x with entries 1, 3, 5, 7, 9. Column 2 is labeled y with entries 3, 7, 11, 15, 19. A 2-column table titled Linear Function B. Column 1 is labeled x with entries 1, 3, 5, 7, 9. Column 2 is labeled y with entries 4, 8, 12, 16, 20.
The initial values of the two functions are different, and the rates of change of the two functions are also different.
The initial values of the two functions are different, and the rates of change of the two functions are the same.
The initial values of the two functions are the same, and the rates of change of the two functions are different.
The initial values of the two functions are the same, and the rates of change of the two functions are also the same.

Answer 1

The correct choice is:
"The initial values of the two functions are different, and the rates of change of the two functions are the same."

Reason: Both tables show y increasing by 4 when x increases by 2, so the slope (rate of change) is 4/2 = 2 for both functions. At x = 1 the outputs are different (A(1)=3, B(1)=4), so the y-intercepts differ. The equations are A(x) = 2x + 1 and B(x) = 2x + 2, so B(x) = A(x) + 1. Because they have the same slope but different intercepts, the outputs differ by the same constant (1) for every input x.