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 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 the first table

For the first table: When $x_1 = 1,y_1=\frac{1}{2}$; $x_2 = 2,y_2 = 1$. The slope $m_1=\frac{1-\frac{1}{2}}{2 - 1}=\frac{\frac{1}{2}}{1}=\frac{1}{2}$. When $x_1 = 2,y_1 = 1$; $x_2 = 3,y_2=1\frac{1}{2}=\frac{3}{2}$. The slope $m_2=\frac{\frac{3}{2}-1}{3 - 2}=\frac{\frac{1}{2}}{1}=\frac{1}{2}$. When $x_1 = 3,y_1=\frac{3}{2}$; $x_2 = 4,y_2 = 2$. The slope $m_3=\frac{2-\frac{3}{2}}{4 - 3}=\frac{\frac{1}{2}}{1}=\frac{1}{2}$. The rate of change is constant.

Step3: Check the second table

For the second table: When $x_1 = 1,y_1 = 1$; $x_2 = 2,y_2=\frac{1}{2}$. The slope $m_1=\frac{\frac{1}{2}-1}{2 - 1}=-\frac{1}{2}$. When $x_1 = 2,y_1=\frac{1}{2}$; $x_2 = 3,y_2=\frac{1}{3}$. The slope $m_2=\frac{\frac{1}{3}-\frac{1}{2}}{3 - 2}=\frac{\frac{2 - 3}{6}}{1}=-\frac{1}{6}$. The rate of change is not constant.

Step4: Check the third table

For the third table: When $x_1 = 1,y_1 = 7$; $x_2 = 2,y_2 = 9$. The slope $m_1=\frac{9 - 7}{2 - 1}=2$. When $x_1 = 2,y_1 = 9$; $x_2 = 3,y_2 = 13$. The slope $m_2=\frac{13 - 9}{3 - 2}=4$. The rate of change is not constant.

Step5: Check the fourth table

For the fourth table: When $x_1 = 1,y_1 = 0$; $x_2 = 2,y_2 = 6$. The slope $m_1=\frac{6 - 0}{2 - 1}=6$. When $x_1 = 2,y_1 = 6$; $x_2 = 3,y_2 = 16$. The slope $m_2=\frac{16 - 6}{3 - 2}=10$. The rate of change is not constant.

Answer:

The first table represents a linear function.