Question

the table shows a function. is the function linear or nonlinear?
x | y
2 | 7\frac{2}{5}
5 | 5\frac{3}{5}
8 | 3\frac{4}{5}
linear nonlinear

Explanation:

Step1: Convert mixed numbers to improper fractions

First, convert the \( y \)-values to improper fractions.

• For \( 7\frac{2}{5} \), we have \( 7\frac{2}{5}=\frac{7\times5 + 2}{5}=\frac{37}{5} \)
• For \( 5\frac{3}{5} \), we have \( 5\frac{3}{5}=\frac{5\times5+3}{5}=\frac{28}{5} \)
• For \( 3\frac{4}{5} \), we have \( 3\frac{4}{5}=\frac{3\times5 + 4}{5}=\frac{19}{5} \)

So the table becomes:

\( x \)	\( y \)
5	\( \frac{28}{5} \)
8	\( \frac{19}{5} \)

Step2: Calculate the rate of change (slope) between consecutive points

The formula for the slope \( m \) between two points \( (x_1,y_1) \) and \( (x_2,y_2) \) is \( m=\frac{y_2 - y_1}{x_2 - x_1} \)

Between \( (2,\frac{37}{5}) \) and \( (5,\frac{28}{5}) \)

\( x_1 = 2,y_1=\frac{37}{5},x_2 = 5,y_2=\frac{28}{5} \)
\( m_1=\frac{\frac{28}{5}-\frac{37}{5}}{5 - 2}=\frac{\frac{28 - 37}{5}}{3}=\frac{\frac{-9}{5}}{3}=\frac{-9}{5}\times\frac{1}{3}=-\frac{3}{5} \)

Between \( (5,\frac{28}{5}) \) and \( (8,\frac{19}{5}) \)

\( x_1 = 5,y_1=\frac{28}{5},x_2 = 8,y_2=\frac{19}{5} \)
\( m_2=\frac{\frac{19}{5}-\frac{28}{5}}{8 - 5}=\frac{\frac{19 - 28}{5}}{3}=\frac{\frac{-9}{5}}{3}=\frac{-9}{5}\times\frac{1}{3}=-\frac{3}{5} \)

Since the slope \( m_1 = m_2=-\frac{3}{5} \) (the rate of change is constant), the function is linear.

Answer:

linear