Question

graphs and functions
identifying linear functions given ordered pairs
for each function, state whether it is linear.
function 1
{(-3, 1), (-1, 4), (1, 7), (3, 10)}
○ linear
○ not linear
function 2
{(-1, 3), (0, 7), (1, 2), (2, -3)}
○ linear
○ not linear
function 3

x	y
5	2
9	6
13	18
17	54

○ linear
○ not linear
function 4

x	y
0	-5
2	-8
4	-9
6	-11

○ linear
○ not linear

Explanation:

Step1: Analyze Function 1

To determine if a function is linear, we check the rate of change (slope) between consecutive points. For Function 1: \(\{(-3, 1), (-1, 4), (1, 7), (3, 10)\}\)

• Slope between \((-3, 1)\) and \((-1, 4)\): \(m_1=\frac{4 - 1}{-1 - (-3)}=\frac{3}{2}\)
• Slope between \((-1, 4)\) and \((1, 7)\): \(m_2=\frac{7 - 4}{1 - (-1)}=\frac{3}{2}\)
• Slope between \((1, 7)\) and \((3, 10)\): \(m_3=\frac{10 - 7}{3 - 1}=\frac{3}{2}\)

All slopes are equal (\(\frac{3}{2}\)), so Function 1 is linear.

Step2: Analyze Function 2

For Function 2: \(\{(-1, 3), (0, 7), (1, 2), (2, -3)\}\)

• Slope between \((-1, 3)\) and \((0, 7)\): \(m_1=\frac{7 - 3}{0 - (-1)} = 4\)
• Slope between \((0, 7)\) and \((1, 2)\): \(m_2=\frac{2 - 7}{1 - 0}=-5\)

Slopes are not equal, so Function 2 is not linear.

Step3: Analyze Function 3

For Function 3 (table: \(x: 5, 9, 13, 17\); \(y: 2, 6, 18, 54\))

• Slope between \((5, 2)\) and \((9, 6)\): \(m_1=\frac{6 - 2}{9 - 5}=1\)
• Slope between \((9, 6)\) and \((13, 18)\): \(m_2=\frac{18 - 6}{13 - 9}=3\)

Slopes are not equal, so Function 3 is not linear.

Step4: Analyze Function 4

For Function 4 (table: \(x: 0, 2, 4, 6\); \(y: -5, -8, -9, -11\))

• Slope between \((0, -5)\) and \((2, -8)\): \(m_1=\frac{-8 - (-5)}{2 - 0}=-\frac{3}{2}\)
• Slope between \((2, -8)\) and \((4, -9)\): \(m_2=\frac{-9 - (-8)}{4 - 2}=-\frac{1}{2}\)

Slopes are not equal, so Function 4 is not linear.

Answer:

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