Question

3)

x	y
-10	9
-5	4
0	2
5	0
10	-3

a. is it a linear function? ____
b. why/why not? ____________________
____________________
____________________
4)

x	y
1	9
3	-4
7	1
11	6
15	11

a. is it a linear function? ____
b. why/why not? ____________________
____________________

Explanation:

Problem 3

Part a

Step1: Recall linear function definition

A linear function has a constant rate of change (slope), calculated as $\frac{\Delta y}{\Delta x}=\frac{y_2 - y_1}{x_2 - x_1}$ between any two points.

Step2: Calculate slopes between points

• Between $(-10,9)$ and $(-5,4)$: $\frac{4 - 9}{-5 - (-10)}=\frac{-5}{5}=-1$
• Between $(-5,4)$ and $(0,2)$: $\frac{2 - 4}{0 - (-5)}=\frac{-2}{5}=-0.4$
• These slopes are not equal, so it's not linear.

Step1: Define slope for linear function

A linear function requires constant $\frac{\Delta y}{\Delta x}$ (slope) between all pairs of points.

Step2: Compare slopes

Calculate slopes between different $(x,y)$ pairs. For $(-10,9)$ & $(-5,4)$, slope is $-1$; for $(-5,4)$ & $(0,2)$, slope is $-0.4$. Slopes differ, so rate of change isn't constant.

Step1: Recall linear function slope rule

Linear functions have constant $\frac{\Delta y}{\Delta x}$ (slope) between any two points.

Step2: Calculate slopes between points

• Between $(1,9)$ and $(3,-4)$: $\frac{-4 - 9}{3 - 1}=\frac{-13}{2}=-6.5$
• Between $(3,-4)$ and $(7,1)$: $\frac{1 - (-4)}{7 - 3}=\frac{5}{4}=1.25$
• Slopes are not equal, so not linear.

Answer:

a. No

Part b