# Average Rate of Change

Find the average rate of change from $1$ to $x$ for the equation $2x^2 + x$. I tried the solving method but it did not help.

• Can you actually show what you did so we can spot the issue you mention? Regards – Amzoti Sep 2 '13 at 1:55

Recall that the definition of the average rate of change of a function $f$ between $x = a$ and $x = b$ is defined to be

$$\frac{f(b) - f(a)}{b - a}$$

Now returning to your example, we have $f(x) = 2x^2 + x$, and $a = 1$, $b = x$. This leads to an average rate of change of

$$\frac{f(x) - f(1)}{x - 1} = \frac{(2x^2 + x) - (2 \cdot 1^2 + 1)}{x - 1} = \frac{2x^2 + x - 3}{x - 1}$$

Now we can simplify this slightly, by noting that

$$2x^2 + x - 3 = (x - 1)(2x + 3)$$

I'll leave the details to you.