Question

graphs and functions
identifying linear functions given ordered pairs
for each function, state whether it is linear.
function 1
{(-4, -1), (-2, -4), (0, -7), (2, -10)}
○ linear
○ not linear
function 2
{(4, 2), (5, 2), (6, 2), (7, 2)}
○ linear
○ not linear
function 3
x | y
-5 | 5
-4 | 6
-3 | 7
-2 | 3
○ linear
○ not linear
function 4
x | y
0 | -2
2 | -5
4 | -10
6 | -14
○ linear
○ not linear

Explanation:

Step1: Analyze Function 1

To check if a function is linear, we calculate the slope between consecutive ordered pairs. The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$.

• Between $(-4, -1)$ and $(-2, -4)$: $m_1=\frac{-4 - (-1)}{-2 - (-4)}=\frac{-3}{2}=-1.5$
• Between $(-2, -4)$ and $(0, -7)$: $m_2=\frac{-7 - (-4)}{0 - (-2)}=\frac{-3}{2}=-1.5$
• Between $(0, -7)$ and $(2, -10)$: $m_3=\frac{-10 - (-7)}{2 - 0}=\frac{-3}{2}=-1.5$

Since the slope is constant, Function 1 is linear.

Step2: Analyze Function 2

For the ordered pairs $\{(4, 2), (5, 2), (6, 2), (7, 2)\}$, calculate the slope:

• Between $(4, 2)$ and $(5, 2)$: $m_1=\frac{2 - 2}{5 - 4}=0$
• Between $(5, 2)$ and $(6, 2)$: $m_2=\frac{2 - 2}{6 - 5}=0$
• Between $(6, 2)$ and $(7, 2)$: $m_3=\frac{2 - 2}{7 - 6}=0$

Constant slope (0), so Function 2 is linear.

Step3: Analyze Function 3

Calculate slopes between consecutive pairs:

• Between $(-5, 5)$ and $(-4, 6)$: $m_1=\frac{6 - 5}{-4 - (-5)}=1$
• Between $(-4, 6)$ and $(-3, 7)$: $m_2=\frac{7 - 6}{-3 - (-4)}=1$
• Between $(-3, 7)$ and $(-2, 3)$: $m_3=\frac{3 - 7}{-2 - (-3)}=\frac{-4}{1}=-4$

Slopes are not constant (1, 1, -4), so Function 3 is not linear.

Step4: Analyze Function 4

Calculate slopes:

• Between $(0, -2)$ and $(2, -5)$: $m_1=\frac{-5 - (-2)}{2 - 0}=\frac{-3}{2}=-1.5$
• Between $(2, -5)$ and $(4, -10)$: $m_2=\frac{-10 - (-5)}{4 - 2}=\frac{-5}{2}=-2.5$

Slopes are not constant (-1.5, -2.5), so Function 4 is not linear.

Answer:

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