Question

the table shows a function. is the function linear or nonlinear?
x | y
-10 | 2
7\frac{1}{2} | 9
10 | 10
linear nonlinear

Explanation:

Step1: Recall linear function property

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

Step2: Calculate slope between first two points

First points: \((x_1,y_1)=(-10,2)\), \((x_2,y_2)=(7\frac{1}{2},9)\). Convert \(7\frac{1}{2}\) to \(7.5\).
Slope \(m_1=\frac{9 - 2}{7.5 - (-10)}=\frac{7}{17.5}=0.4\).

Step3: Calculate slope between last two points

Points: \((x_2,y_2)=(7.5,9)\), \((x_3,y_3)=(10,10)\).
Slope \(m_2=\frac{10 - 9}{10 - 7.5}=\frac{1}{2.5}=0.4\).

Step4: Compare slopes

Since \(m_1 = m_2 = 0.4\), the rate of change is constant.

Answer:

linear