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, -2), (6, -5)}
○ linear
○ not linear
function 2
{(-2, 2), (-1, 4), (0, 8), (1, 13)}
○ linear
○ not linear
function 3

x	y
-4	-1
-2	-6
0	-1
2	4

○ linear
○ not linear
function 4

x	y
4	-3
8	-3
12	-3
16	-3

○ linear
○ not linear

Explanation:

Step1: Analyze Function 1

To determine if a function is linear, we check the rate of change (slope) between consecutive ordered pairs. For Function 1: \((3, 4)\), \((4, 1)\), \((5, -2)\), \((6, -5)\).

• Slope between \((3, 4)\) and \((4, 1)\): \(m_1=\frac{1 - 4}{4 - 3}=\frac{-3}{1}=-3\)
• Slope between \((4, 1)\) and \((5, -2)\): \(m_2=\frac{-2 - 1}{5 - 4}=\frac{-3}{1}=-3\)
• Slope between \((5, -2)\) and \((6, -5)\): \(m_3=\frac{-5 - (-2)}{6 - 5}=\frac{-3}{1}=-3\)

All slopes are equal, so Function 1 is linear.

Step2: Analyze Function 2

For Function 2: \((-2, 2)\), \((-1, 4)\), \((0, 8)\), \((1, 13)\).

• Slope between \((-2, 2)\) and \((-1, 4)\): \(m_1=\frac{4 - 2}{-1 - (-2)}=\frac{2}{1}=2\)
• Slope between \((-1, 4)\) and \((0, 8)\): \(m_2=\frac{8 - 4}{0 - (-1)}=\frac{4}{1}=4\)
• Slope between \((0, 8)\) and \((1, 13)\): \(m_3=\frac{13 - 8}{1 - 0}=\frac{5}{1}=5\)

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

Step3: Analyze Function 3

For Function 3 (table): \(x: - 4, - 2, 0, 2\); \(y: - 1, - 6, - 1, 4\).

• Slope between \((-4, -1)\) and \((-2, -6)\): \(m_1=\frac{-6 - (-1)}{-2 - (-4)}=\frac{-5}{2}=-2.5\)
• Slope between \((-2, -6)\) and \((0, -1)\): \(m_2=\frac{-1 - (-6)}{0 - (-2)}=\frac{5}{2}=2.5\)

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

Step4: Analyze Function 4

For Function 4 (table): \(x: 4, 8, 12, 16\); \(y: - 3, - 3, - 3, - 3\).
The \(y\)-value is constant (\(y = - 3\)) for all \(x\)-values. The slope (rate of change) is \(\frac{-3 - (-3)}{x_2 - x_1}=0\) for any two points. So it is a horizontal line, hence linear.

Answer:

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