What is the average rate of change of $f(x)$ from $x_{1}=-7$ to $x_{2}=-3$ ? Please write your answer rounded to the nearest hundredth. \[ f(x)=\sqrt{-7 x+2} \]

Step 1 ：Given the function \(f(x)=\sqrt{-7 x+2}\), we are asked to find the average rate of change from \(x_{1}=-7\) to \(x_{2}=-3\).

Step 2 ：The average rate of change of a function \(f(x)\) from \(x_{1}\) to \(x_{2}\) is given by the formula: \[\frac{f(x_{2}) - f(x_{1})}{x_{2} - x_{1}}\]

Step 3 ：First, we substitute \(x_{1}=-7\) and \(x_{2}=-3\) into the function \(f(x)=\sqrt{-7 x+2}\) to find the values of \(f(x_{1})\) and \(f(x_{2})\).

Step 4 ：We find that \(f(x_{1}) = 7.14142842854285\) and \(f(x_{2}) = 4.795831523312719\).

Step 5 ：Next, we substitute these values into the formula for the average rate of change: \[\frac{f(x_{2}) - f(x_{1})}{x_{2} - x_{1}} = \frac{4.795831523312719 - 7.14142842854285}{-3 - (-7)}\]

Step 6 ：Solving this gives us an average rate of change of -0.5863992263075328.

Step 7 ：Rounding to the nearest hundredth, we find that the average rate of change of \(f(x)\) from \(x_{1}=-7\) to \(x_{2}=-3\) is \(\boxed{-0.59}\).