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The formula for the rate of change over the interval \(x = a\) to \(x = b\) is $$\frac{f(b) - f(a)}{b - a} $$
These problems are quite formulaic (literally). Just take the points, put them into the formula, and solve
If \(f(x) = x^2 -3\), find the average rate of change between \(x = 2\) and \(x=5\)
Part 1:
Since two is the first \(x\) value, \(a = 2\), and \(b = 5\). Plugging this into the formula, we get
$$\frac{f(5) - f(2)}{5 - 2}$$
\(f(5)\) is \(22\), and \(f(2)\) is \(1\), so we get
$$\frac{22 - 1}{3}$$
This is \(7\), which is the average rate of change
Note that the average rate of change can be negative, which means that the function is decreasing
If \(f(x) = x^2 -4x+2\), find the average rate of change between \(x = 1\) and \(x=3\)
Since two is the first \(x\) value, \(a = 0\), and \(b = 1\). Plugging this into the formula, we get $$\frac{f(1) - f(0)}{1 - 0}$$ \(f(1)\) is \(-1\), and \(f(0)\) is \(2\), so we get $$\frac{-1 - 2}{1-0}$$ This is \(-3\), which is the average rate of change
Conlcusion:
You have now learned how to find the average rate of change of a function between two values of \(x\).
Next: Forms of a line