Final answer:
Options a. {(+6,6), (-3,3), (0,0), (3, 3), (6,6)} and c. {(-2,6), (-1,3), (0,0), (1, -3), (2, -6)} both represent linear functions because they display a constant rate of change. Option a. has a slope of 1 (y=x), and Option c. has a negative slope where y decreases by 3 for every increase of 1 in x, making them linear.
Step-by-step explanation:
To determine which set represents a linear function, we must remember that a linear function is represented by a straight line when graphed, and its equation is of the form y = mx + b, where 'm' is the slope and 'b' is the y-intercept. A linear function has a constant rate of change, meaning that for a constant change in 'x', there should be a constant change in 'y'.
Let's analyze each option:
Option a. has equal increases in 'x' corresponding to equal increases in 'y', which indicates it could be a linear function with a slope of 1 (as y=x for all given points).
Option b. doesn't have a constant relationship between the changes in 'x' and changes in 'y', meaning it's not linear.
Option c. also shows a constant rate of change, with each increase in 'x' by 1 leading to a decrease in 'y' by 3, suggesting it might be a linear function as well with a negative slope.
Option d. has all points with the same 'x'-coordinate, which indicates this is a vertical line, not a function since it fails the vertical line test.
Therefore, the options that represent linear functions are Option a. and Option c., as they both show a constant rate of change and would be represented by a straight line when graphed.