Question

graphs and functions
identifying linear functions given ordered pairs
for each function, state whether it is linear.
function 1
{(2, -5), (3, -8), (4, -11), (5, -14)}
○ linear
○ not linear
function 2
{(-2, 4), (2, 5), (6, 8), (10, 12)}
○ linear
○ not linear
function 3

x	y
-1	-1
2	0
5	1
8	2

○ linear
○ not linear
function 4

x	y
3	0
5	0
7	0
9	0

○ linear
○ not linear

Explanation:

Step1: Analyze Function 1

For a function to be linear, the slope between consecutive points should be constant. The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$.

• Between $(2, -5)$ and $(3, -8)$: $m_1 = \frac{-8 - (-5)}{3 - 2} = \frac{-3}{1} = -3$
• Between $(3, -8)$ and $(4, -11)$: $m_2 = \frac{-11 - (-8)}{4 - 3} = \frac{-3}{1} = -3$
• Between $(4, -11)$ and $(5, -14)$: $m_3 = \frac{-14 - (-11)}{5 - 4} = \frac{-3}{1} = -3$

Since all slopes are equal, Function 1 is linear.

Step2: Analyze Function 2

Calculate slopes between consecutive points.

• Between $(-2, 4)$ and $(2, 5)$: $m_1 = \frac{5 - 4}{2 - (-2)} = \frac{1}{4} = 0.25$
• Between $(2, 5)$ and $(6, 8)$: $m_2 = \frac{8 - 5}{6 - 2} = \frac{3}{4} = 0.75$
• Between $(6, 8)$ and $(10, 12)$: $m_3 = \frac{12 - 8}{10 - 6} = \frac{4}{4} = 1$

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

Step3: Analyze Function 3

Calculate slopes between consecutive points.

• Between $(-1, -1)$ and $(2, 0)$: $m_1 = \frac{0 - (-1)}{2 - (-1)} = \frac{1}{3} \approx 0.333$
• Between $(2, 0)$ and $(5, 1)$: $m_2 = \frac{1 - 0}{5 - 2} = \frac{1}{3} \approx 0.333$
• Between $(5, 1)$ and $(8, 2)$: $m_3 = \frac{2 - 1}{8 - 5} = \frac{1}{3} \approx 0.333$

Slopes are equal, so Function 3 is linear.

Step4: Analyze Function 4

All $y$-values are 0. The slope between any two points is $m = \frac{0 - 0}{x_2 - x_1} = 0$, which is constant. So Function 4 is linear.

Answer:

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