Using Calculus to Sketch Graphs of Polynomial Functions.

This next section you will develop skills in using calculus to find the major characteristics of a function. You already know how to find the x intercepts (by setting the equation equal to zero ie making y=0 ) and how to find the y intercept (by setting x=0). That was helpful, but calculus now allows you to find what are called “stationary points”. These are places on the graph of a function where the gradient is equal to zero. That is f'(x) (or dy/dx) is zero. For example if we have f(x)= x^2 + 6, then f'(x)= 2x. If we know that there is a stationary point (turning point in this case) where f'(x)=0, the we can let 2x=0 and find the x value where this occurs (in this case x=0). So we know that the graph will turn at x=0. From here we can sketch the graph. More is involved and there are several types of stationary points (local maximum, local minimum and points of inflection) but calculus lets you investigate these and extract valuable information that lets you sketch the graph fairly painlessly. On the Wiki is the beginning of Ex 12.2.

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