Cubic Graphs

I think I understand cubic graphs except for one thing: How do I know whether the graph crosses a point or touches it? I know how to get the points that the graph will cross or touch, but I do not know how to determine whether it will cross or touch the point in question. Could anyone elaborate?

Ignore previous post. The graph will touch the x-axis if 2 or 3 of the roots are the same. For example, the cubic graph (x+1)(x+2)^2 will cross the x-axis at -1 and touch the point -2, not go through it.

I'm only going on what my textbook tells me. Sometimes, the graph will dip under the axis after crossing the point, and sometimes, in some of the graphs, the graph will 'touch' the point, not cross over the axis, and continue going up or down without crossing over.

It still goes THROUGH $(-2,0)$...



I think what you mean is 'How do I determine whether the x-axis will be tangent to the cubic?' in which case, you should know that a cubic will ALWAYS have 1 real root, ie it will always cross the x-axis at some point. If you know that point, you can divide your cubic through by $(x-a)$ where a is that point, and see how many roots the leftover quadratic has. If 0, then the cubic does not touch the x-axis (excluding the point $a$), if 1 then the cubic touches the x-axis once, and if 2 then the cubic intersects the x-axis two more times.

Also if you can factorise your cubic such that $(x-a)(x-b)(x-c)=0$ then if any two pairs of $a,b,c$ are equal to each other, and the other is distinct, then you'll have the situation where it does touch the x-axis.