Question

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

x	y
$1\frac{1}{5}$	9
$5\frac{1}{5}$	4
6	2

options: linear, nonlinear

Explanation:

Step1: Recall linear function definition

A linear function has a constant rate of change (slope), calculated as $\frac{\Delta y}{\Delta x}$ between consecutive points.

Step2: Convert mixed numbers to improper fractions

First point: $x_1 = 1\frac{1}{5}=\frac{6}{5}$, $y_1 = 9$
Second point: $x_2 = 5\frac{1}{5}=\frac{26}{5}$, $y_2 = 4$
Third point: $x_3 = 6=\frac{30}{5}$, $y_3 = 2$

Step3: Calculate slope between first and second point

$\Delta x_1 = x_2 - x_1=\frac{26}{5}-\frac{6}{5}=\frac{20}{5}=4$
$\Delta y_1 = y_2 - y_1 = 4 - 9=-5$
Slope $m_1=\frac{\Delta y_1}{\Delta x_1}=\frac{-5}{4}=-\frac{5}{4}$

Step4: Calculate slope between second and third point

$\Delta x_2 = x_3 - x_2=\frac{30}{5}-\frac{26}{5}=\frac{4}{5}$
$\Delta y_2 = y_3 - y_2 = 2 - 4=-2$
Slope $m_2=\frac{\Delta y_2}{\Delta x_2}=\frac{-2}{\frac{4}{5}}=-2\times\frac{5}{4}=-\frac{5}{2}$

Step5: Compare slopes

Since $m_1 = -\frac{5}{4}$ and $m_2 = -\frac{5}{2}$ are not equal, the rate of change is not constant.

Answer:

nonlinear