Question

the table below represents a linear function. identify the rate of change of the function.
(table with x and y values: x=1, y=8; x=4, y=6; x=7, y=4; x=10, y=2)

Explanation:

Step1: Recall the slope formula

The rate of change (slope) of a linear function is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \), where \((x_1,y_1)\) and \((x_2,y_2)\) are two points on the line.

Step2: Select two points from the table

Let's take the first two points \((x_1,y_1)=(1,1)\) and \((x_2,y_2)=(4,8)\).

Step3: Calculate the slope

Substitute into the slope formula: \( m=\frac{8 - 1}{4 - 1}=\frac{7}{3}\approx2.33\). Wait, maybe I misread the table. Let's check another pair. Let's take \((1,1)\) and \((7,15)\). Then \( m=\frac{15 - 1}{7 - 1}=\frac{14}{6}=\frac{7}{3}\approx2.33\). Another pair: \((4,8)\) and \((7,15)\), \( m=\frac{15 - 8}{7 - 4}=\frac{7}{3}\approx2.33\). And \((7,15)\) and \((10,22)\), \( m=\frac{22 - 15}{10 - 7}=\frac{7}{3}\approx2.33\). So the rate of change is \(\frac{7}{3}\) or approximately \(2.33\).

Answer:

The rate of change of the linear function is \(\frac{7}{3}\) (or approximately \(2.33\)).