Question

4)

x	y
3	-4
7	1
11	6
15	11

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

Explanation:

Part a

Step1: Recall linear function definition

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

Step2: Calculate slope between first two points

Take \((x_1,y_1)=(1,9)\) and \((x_2,y_2)=(3, - 4)\).
Slope \( m_1=\frac{-4 - 9}{3 - 1}=\frac{-13}{2}=-6.5 \).

Step3: Calculate slope between next two points

Take \((x_2,y_2)=(3, - 4)\) and \((x_3,y_3)=(7,1)\).
Slope \( m_2=\frac{1-(-4)}{7 - 3}=\frac{5}{4} = 1.25 \).

Step4: Compare slopes

Since \( m_1=-6.5
eq m_2 = 1.25 \), the rate of change is not constant.

A linear function requires a constant slope (rate of change) between all pairs of points. We calculated the slope between \((1,9)\) and \((3, - 4)\) as \(-\frac{13}{2}\) and between \((3, - 4)\) and \((7,1)\) as \(\frac{5}{4}\). These slopes are not equal, so the function does not have a constant rate of change, meaning it is not linear.

Answer:

No

Part b