Question

which set of ordered pairs (x,y) could represent a linear function? a = {(-1,4), (1,2), (3,-1), (5,-3)} b = {(-5,6), (-1,1), (1,-2), (3,-5)} c = {(-4,-7), (-2,-4), (0,-1), (2,2)} d = {(2,8), (3,5), (4,3), (5,0)}

Explanation:

Step1: Recall the property of linear functions

A linear function has a constant slope between any two points. The slope \( m \) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is calculated by \( m=\frac{y_2 - y_1}{x_2 - x_1} \). We will calculate the slope between consecutive points for each set.

Step2: Calculate slope for set A

• Between \((-1,4)\) and \((1,2)\): \( m_1=\frac{2 - 4}{1-(-1)}=\frac{-2}{2}=- 1 \)
• Between \((1,2)\) and \((3,-1)\): \( m_2=\frac{-1 - 2}{3 - 1}=\frac{-3}{2}=-1.5 \)

Since \( m_1
eq m_2 \), set A is not linear.

Step3: Calculate slope for set B

• Between \((-5,6)\) and \((-1,1)\): \( m_1=\frac{1 - 6}{-1-(-5)}=\frac{-5}{4}=-1.25 \)
• Between \((-1,1)\) and \((1,-2)\): \( m_2=\frac{-2 - 1}{1-(-1)}=\frac{-3}{2}=-1.5 \)

Since \( m_1
eq m_2 \), set B is not linear.

Step4: Calculate slope for set C

• Between \((-4,-7)\) and \((-2,-4)\): \( m_1=\frac{-4-(-7)}{-2 - (-4)}=\frac{3}{2}=1.5 \)
• Between \((-2,-4)\) and \((0,-1)\): \( m_2=\frac{-1-(-4)}{0 - (-2)}=\frac{3}{2}=1.5 \)
• Between \((0,-1)\) and \((2,2)\): \( m_3=\frac{2-(-1)}{2 - 0}=\frac{3}{2}=1.5 \)

All slopes are equal, so set C has a constant slope.

Step5: Calculate slope for set D (optional, but for completeness)

• Between \((2,8)\) and \((3,5)\): \( m_1=\frac{5 - 8}{3 - 2}=-3 \)
• Between \((3,5)\) and \((4,3)\): \( m_2=\frac{3 - 5}{4 - 3}=-2 \)

Since \( m_1
eq m_2 \), set D is not linear.

Answer:

C. \(\{(-4, -7), (-2, -4), (0, -1), (2, 2)\}\)