which table represents a linear function?\nmark this and return

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 ((1,5)), ((2,9)), ((3,5)), ((4,9)): The slope between ((1,5)) and ((2,9)) is (m_1=\frac{9 - 5}{2 - 1}=4). The slope between ((2,9)) and ((3,5)) is (m_2=\frac{5 - 9}{3 - 2}=-4). Since (m_1\neq m_2), it is not a linear function.

Step3: Check the second table

For the second table with points ((1, - 5)), ((2,10)), ((3,-15)), ((4,20)): The slope between ((1,-5)) and ((2,10)) is (m_1=\frac{10+5}{2 - 1}=15). The slope between ((2,10)) and ((3,-15)) is (m_2=\frac{-15 - 10}{3 - 2}=-25). Since (m_1\neq m_2), it is not a linear function.

Step4: Check the third table

For the third table with points ((1,5)), ((2,10)), ((3,20)), ((4,40)): The slope between ((1,5)) and ((2,10)) is (m_1=\frac{10 - 5}{2 - 1}=5). The slope between ((2,10)) and ((3,20)) is (m_2=\frac{20 - 10}{3 - 2}=10). Since (m_1\neq m_2), it is not a linear function.

Step5: Check the fourth table

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

Answer:

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