Write an article based on this "Know your function. Select values of x. Calculate the values of the function. Calculate the average rate of change of the function. Interpret the result."
In mathematics, a function is a mathematical relationship between numbers, so that you enter one number and another number is the result. Functions can generally be graphed. They may represents straight lines, parabolas, or random-looking curves that have no easy definition. Some sample functions are:   y(x)=3x+4{\displaystyle y(x)=3x+4} (the function of a straight line)  y(x)=sin(x){\displaystyle y(x)=sin(x)} (a function for a waved line)  y(x)=x2{\displaystyle y(x)=x^{2}} (a function for a parabola) Finding the average rate of change of a function means measuring the value of the function at two different points along the x-axis. Select one value of x where you wish to begin measuring, and then determine how far along the axis you wish to advance. Depending on your purposes, you may choose a wider or narrower range of x values to measure. For this exercise, select the first x-value at 0 and the second x-value at 3. Rate of change of the function measures how much the y-values change over the chosen horizontal x-distance. To calculate this change, you need to know the y values at each chosen value of x. For the sample function, y(x)=x2{\displaystyle y(x)=x^{2}}, select the two values x=0 and x=3, for example. The corresponding values of y(x){\displaystyle y(x)}, therefore, are:  y(0)=02=0{\displaystyle y(0)=0^{2}=0} y(3)=32=9{\displaystyle y(3)=3^{2}=9} The rate of change of a function can be written formally as:  A(x)=ΔyΔx=f(x+h)−f(x)h{\displaystyle A(x)={\frac {\Delta y}{\Delta x}}={\frac {f(x+h)-f(x)}{h}}} In this formula, f(x){\displaystyle f(x)} represents the value of the function at the first chosen x-value. f(x+h){\displaystyle f(x+h)} is the value of the function some distance away at a second value of x. The denominator h{\displaystyle h} is the distance between the two measurements.  h{\displaystyle h} can also be represented as Δx{\displaystyle \Delta x}, since it is the distance or change in the chosen x-values. For the chosen function y(x)=x2{\displaystyle y(x)=x^{2}}, you can calculate the average rate of change from 0 to 3 as follows:  A(x)=ΔyΔx=9−03−0=3{\displaystyle A(x)={\frac {\Delta y}{\Delta x}}={\frac {9-0}{3-0}}=3}. For this function, the rate of change is a measure of how much the value of the function changes vertically as you move horizontally along the x-axis. In this case, the parabola y(x)=x2{\displaystyle y(x)=x^{2}} begins at point (0,0) and climbs to point (3,9) over the measured interval. Although the function itself is not a straight line, the average rate of change is measured as the slope of the straight line connecting those two points. This line climbs 3 units for each single unit increase in x. Note that the average rate of change for a function may differ depending on the location that you choose to measure. For the parabola example, the average rate of change is 3 from x=0 to x=3. However, for the same function measured from x=3 to x=6, also a distance of 3 units, the average rate of change becomes 8.33.