Question

graphs and functions
identifying linear functions given ordered pairs
for each function, state whether it is linear.
function 1
{(-3, 2), (0, 4), (3, 6), (6, 8)}
linear
not linear
function 2
{(4, -1), (8, -3), (12, -4), (16, -7)}
linear
not linear
function 3
x | y
2 | 0
3 | 0
4 | 0
5 | 0
linear
not linear
function 4
x | y
-5 | -3
-1 | -4
3 | -5
7 | -6
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. The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$.
For Function 1: Pairs $(-3, 2)$ and $(0, 4)$: $m_1=\frac{4 - 2}{0 - (-3)}=\frac{2}{3}$.
Pairs $(0, 4)$ and $(3, 6)$: $m_2=\frac{6 - 4}{3 - 0}=\frac{2}{3}$.
Pairs $(3, 6)$ and $(6, 8)$: $m_3=\frac{8 - 6}{6 - 3}=\frac{2}{3}$.
Since all slopes are equal, Function 1 is linear.

Step2: Analyze Function 2

Pairs $(4, -1)$ and $(8, -3)$: $m_1=\frac{-3 - (-1)}{8 - 4}=\frac{-2}{4}=-\frac{1}{2}$.
Pairs $(8, -3)$ and $(12, -4)$: $m_2=\frac{-4 - (-3)}{12 - 8}=\frac{-1}{4}$.
Since $m_1
eq m_2$, Function 2 is not linear.

Step3: Analyze Function 3

The $y$-value is always 0 for different $x$-values. The slope between any two points: e.g., $(2, 0)$ and $(3, 0)$: $m=\frac{0 - 0}{3 - 2}=0$. All slopes are 0, so Function 3 is linear.

Step4: Analyze Function 4

Pairs $(-5, -3)$ and $(-1, -4)$: $m_1=\frac{-4 - (-3)}{-1 - (-5)}=\frac{-1}{4}$.
Pairs $(-1, -4)$ and $(3, -5)$: $m_2=\frac{-5 - (-4)}{3 - (-1)}=\frac{-1}{4}$.
Pairs $(3, -5)$ and $(7, -6)$: $m_3=\frac{-6 - (-5)}{7 - 3}=\frac{-1}{4}$.
All slopes are equal, so Function 4 is linear.

Answer:

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