Question

functions and lines
identifying linear functions given ordered pairs
for each function, state whether it is linear.
function 1
{(3, -4), (4, 1), (5, 6), (6, 11)}
○ linear
○ not linear
function 2
{(-5, 0), (-3, -1), (-1, -2), (1, -3)}
○ linear
○ not linear
function 3

x	y
4	1
8	1
12	1
16	1

○ linear
○ not linear
function 4

x	y
-4	-1
-3	-3
-2	-4
-1	-7

○ linear
○ not linear

Explanation:

Function 1:

Step1: Calculate the slope between consecutive points.

For points \((3, -4)\) and \((4, 1)\): \(m_1=\frac{1 - (-4)}{4 - 3}=\frac{5}{1} = 5\)
For points \((4, 1)\) and \((5, 6)\): \(m_2=\frac{6 - 1}{5 - 4}=\frac{5}{1}=5\)
For points \((5, 6)\) and \((6, 11)\): \(m_3=\frac{11 - 6}{6 - 5}=\frac{5}{1}=5\)

Step2: Check if slopes are equal.

All slopes are \(5\), so Function 1 is linear.

Function 2:

Step1: Calculate the slope between consecutive points.

For points \((-5, 0)\) and \((-3, -1)\): \(m_1=\frac{-1 - 0}{-3 - (-5)}=\frac{-1}{2}=-0.5\)
For points \((-3, -1)\) and \((-1, -2)\): \(m_2=\frac{-2 - (-1)}{-1 - (-3)}=\frac{-1}{2}=-0.5\)
For points \((-1, -2)\) and \((1, -3)\): \(m_3=\frac{-3 - (-2)}{1 - (-1)}=\frac{-1}{2}=-0.5\)

Step2: Check if slopes are equal.

All slopes are \(-0.5\), so Function 2 is linear.

Function 3:

Step1: Calculate the slope between consecutive points.

For points \((4, 1)\) and \((8, 1)\): \(m_1=\frac{1 - 1}{8 - 4}=\frac{0}{4}=0\)
For points \((8, 1)\) and \((12, 1)\): \(m_2=\frac{1 - 1}{12 - 8}=\frac{0}{4}=0\)
For points \((12, 1)\) and \((16, 1)\): \(m_3=\frac{1 - 1}{16 - 12}=\frac{0}{4}=0\)

Step2: Check if slopes are equal.

All slopes are \(0\), so Function 3 is linear (horizontal line).

Function 4:

Step1: Calculate the slope between consecutive points.

For points \((-4, -1)\) and \((-3, -3)\): \(m_1=\frac{-3 - (-1)}{-3 - (-4)}=\frac{-2}{1}=-2\)
For points \((-3, -3)\) and \((-2, -4)\): \(m_2=\frac{-4 - (-3)}{-2 - (-3)}=\frac{-1}{1}=-1\)

Step2: Check if slopes are equal.

Slopes \(m_1=-2\) and \(m_2 = -1\) are not equal, so Function 4 is not linear.

Answer:

Function 1: Linear
Function 2: Linear
Function 3: Linear
Function 4: Not linear