which table represents a linear function?

Answer:

The second table (with (x = 0,y = 3;x = 1,y = 1;x = 2,y=-1;x = 3,y = - 3)) represents a linear function.

Explanation:

Step1: Recall linear - function property

A linear function has a constant rate of change (slope). The slope (m=\frac{\Delta y}{\Delta x}=\frac{y_2 - y_1}{x_2 - x_1}).

Step2: Check first table

For (x = 0,y = 1) and (x = 1,y = 3), (m_1=\frac{3 - 1}{1-0}=2). For (x = 1,y = 3) and (x = 2,y=-3), (m_2=\frac{-3 - 3}{2 - 1}=-6). Since (m_1\neq m_2), it's not linear.

Step3: Check second table

For (x = 0,y = 3) and (x = 1,y = 1), (m_1=\frac{1 - 3}{1-0}=-2). For (x = 1,y = 1) and (x = 2,y=-1), (m_2=\frac{-1 - 1}{2 - 1}=-2). For (x = 2,y=-1) and (x = 3,y=-3), (m_3=\frac{-3+1}{3 - 2}=-2). Since the slope is constant ((m=-2)), it's linear.

Step4: Check third table

For (x = 0,y = 0) and (x = 1,y = 1), (m_1=\frac{1 - 0}{1-0}=1). For (x = 1,y = 1) and (x = 2,y = 4), (m_2=\frac{4 - 1}{2 - 1}=3). Since (m_1\neq m_2), it's not linear.

Step5: Check fourth table

For (x = 0,y = 0) and (x = 1,y=-1), (m_1=\frac{-1 - 0}{1-0}=-1). For (x = 1,y=-1) and (x = 2,y = 4), (m_2=\frac{4 + 1}{2 - 1}=5). Since (m_1\neq m_2), it's not linear.