How to find the nature of the stationary point and sketch the graph?

You use dy/dx to find the gradient, right? Well, to find the nature of a stationary point, you first have to have the stationary point itself, that is when dy/dx is 0. Then, find d^2y/dx^2 (\frac{d^2y}{dx^2}). Is the value of this is negative, then the point is a maximum. If it is a positive value, the point is a minimum. If you obtain 0 when substituting x by the x coordinate of the stationary point, then the point is an inflexion.

To draw the graph, it is more complicated, but easy if you know the basic shapes and how to manipulate with them. Once you have the turning point, plot it. Then, draw a curve according to its nature, with the turning point as reference. Set y to zero to find the x-intercepts, and x to zero to find the y intercept.

Hope it helped! Just post if you want more details.

Interesting. We never used the terminology "stationary point".Lol, we use 'stationary point', 'turning point', 'maximum' or 'minimum' or 'optimum point'. And all means to me a point on a curve where gradient is zero.