# What is the average rate of change of the function y = 2 x^2 + 3 between x = x_1 and x = x_2?

## Question:

What is the average rate of change of the function {eq}\displaystyle y = 2 x^2 + 3 {/eq} between {eq}x = x_1 {/eq} and {eq}x = x_2 {/eq}?

## Slope

The slope of a function is given by the difference of the value of the function at both points divided by the difference of both points. It is denoted by m.

Given,

{eq}\begin{align*} f\left( x \right) = 2{x^2} + 3\\ \end{align*} {/eq}

We have to find average rate of change when

{eq}\begin{align*} x = {x_1}\;\\ x = {x_2} \end{align*} {/eq}

We know the slope m is

{eq}\begin{align*} m &= \dfrac{{f\left( {{x_2}} \right) - f\left( {{x_1}} \right)}}{{{x_2} - {x_1}}}\\ &= \dfrac{{2{{\left( {{x_2}} \right)}^2} + 3 - 2{{\left( {{x_1}} \right)}^2} - 3}}{{{x_2} - {x_1}}}\\ &= \dfrac{{2\left( {{{\left( {{x_2}} \right)}^2} - {{\left( {{x_1}} \right)}^2}} \right)}}{{{x_2} - {x_1}}}\\ &= \dfrac{{2\left( {{x_2} - {x_1}} \right)\left( {{x_2} + {x_1}} \right)}}{{{x_2} - {x_1}}}\\ m &= 2\left( {{x_2} + {x_1}} \right) \end{align*} {/eq}

This slope is the average rate of change of the above function.