determining if a function is linear\nwhich table represents a linear function?

Explanation:

Step1: Recall linear - function property

A linear function has a constant rate of change (slope). The slope formula between two points ((x_1,y_1)) and ((x_2,y_2)) is (m=\frac{y_2 - y_1}{x_2 - x_1}).

Step2: Check the first table

For the first table with points ((0,1)), ((1,2)), ((2,4)), ((3,8)): Between ((0,1)) and ((1,2)), (m_1=\frac{2 - 1}{1 - 0}=1). Between ((1,2)) and ((2,4)), (m_2=\frac{4 - 2}{2 - 1}=2). Since (m_1\neq m_2), it is not linear.

Step3: Check the second table

For the second table with points ((0,0)), ((1,1)), ((2,3)), ((3,6)): Between ((0,0)) and ((1,1)), (m_1=\frac{1 - 0}{1 - 0}=1). Between ((1,1)) and ((2,3)), (m_2=\frac{3 - 1}{2 - 1}=2). Since (m_1\neq m_2), it is not linear.

Step4: Check the third table

For the third table with points ((0,0)), ((1,1)), ((2,0)), ((3,1)): Between ((0,0)) and ((1,1)), (m_1=\frac{1 - 0}{1 - 0}=1). Between ((1,1)) and ((2,0)), (m_2=\frac{0 - 1}{2 - 1}=- 1). Since (m_1\neq m_2), it is not linear.

Step5: Check the fourth table

For the fourth table with points ((0,1)), ((1,3)), ((2,5)), ((3,7)): Between ((0,1)) and ((1,3)), (m_1=\frac{3 - 1}{1 - 0}=2). Between ((1,3)) and ((2,5)), (m_2=\frac{5 - 3}{2 - 1}=2). Between ((2,5)) and ((3,7)), (m_3=\frac{7 - 5}{3 - 2}=2). Since the slope is constant ((m = 2)), it is a linear function.

Answer:

The fourth table (with (x = 0,y = 1); (x = 1,y = 3); (x = 2,y = 5); (x = 3,y = 7)) represents a linear function.