Question

the table shows a function. is the function linear or nonlinear?

x	y
-10	0
2	-3
8	-4\frac{1}{2}

options: linear, nonlinear

Explanation:

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

First points: \((x_1,y_1)=(-10,0)\), \((x_2,y_2)=(2,-3)\).
Slope \( m_1=\frac{-3 - 0}{2 - (-10)}=\frac{-3}{12}=-\frac{1}{4} \).

Step3: Calculate slope between second and third points

Second points: \((x_2,y_2)=(2,-3)\), \((x_3,y_3)=(8,-4\frac{1}{2})\) (convert \( -4\frac{1}{2} \) to \( -\frac{9}{2} \)).
Slope \( m_2=\frac{-\frac{9}{2}-(-3)}{8 - 2}=\frac{-\frac{9}{2}+\frac{6}{2}}{6}=\frac{-\frac{3}{2}}{6}=-\frac{3}{2}\times\frac{1}{6}=-\frac{1}{4} \).

Step4: Compare slopes

Since \( m_1 = m_2=-\frac{1}{4} \), the rate of change is constant.

Answer:

linear