which table represents a linear function?\n| x | y |\n|----|----|\n| 1 | 1/2 |\n| 2 | 1 |\n| 3 | 1 1/2 |\n|…

Explanation:

Step1: Recall slope - formula

The slope formula for a linear function between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$. For a linear function, the slope between any two points is constant.

Step2: Check the first table

For the first table with points $(1,\frac{1}{2}),(2,1),(3,\frac{3}{2}),(4,2)$: When $x_1 = 1,y_1=\frac{1}{2},x_2 = 2,y_2 = 1$, $m_1=\frac{1-\frac{1}{2}}{2 - 1}=\frac{\frac{1}{2}}{1}=\frac{1}{2}$. When $x_1 = 2,y_1 = 1,x_2=3,y_2=\frac{3}{2}$, $m_2=\frac{\frac{3}{2}-1}{3 - 2}=\frac{\frac{1}{2}}{1}=\frac{1}{2}$. When $x_1 = 3,y_1=\frac{3}{2},x_2 = 4,y_2 = 2$, $m_3=\frac{2-\frac{3}{2}}{4 - 3}=\frac{\frac{1}{2}}{1}=\frac{1}{2}$. The slope is constant.

Step3: Check the second table

For the second table with points $(1,1),(2,\frac{1}{2}),(3,\frac{1}{3}),(4,\frac{1}{4})$: When $x_1 = 1,y_1 = 1,x_2 = 2,y_2=\frac{1}{2}$, $m_1=\frac{\frac{1}{2}-1}{2 - 1}=\frac{-\frac{1}{2}}{1}=-\frac{1}{2}$. When $x_1 = 2,y_1=\frac{1}{2},x_2 = 3,y_2=\frac{1}{3}$, $m_2=\frac{\frac{1}{3}-\frac{1}{2}}{3 - 2}=\frac{\frac{2 - 3}{6}}{1}=-\frac{1}{6}$. The slope is not constant.

Step4: Check the third table

For the third table with points $(1,7),(2,9),(3,13),(4,21)$: When $x_1 = 1,y_1 = 7,x_2 = 2,y_2 = 9$, $m_1=\frac{9 - 7}{2 - 1}=2$. When $x_1 = 2,y_1 = 9,x_2 = 3,y_2 = 13$, $m_2=\frac{13 - 9}{3 - 2}=4$. The slope is not constant.

Step5: Check the fourth table

For the fourth table with points $(1,0),(2,6),(3,16),(4,30)$: When $x_1 = 1,y_1 = 0,x_2 = 2,y_2 = 6$, $m_1=\frac{6 - 0}{2 - 1}=6$. When $x_1 = 2,y_1 = 6,x_2 = 3,y_2 = 16$, $m_2=\frac{16 - 6}{3 - 2}=10$. The slope is not constant.

Answer:

The first table represents a linear function.