Points of Intersection

In order to find the points of intersection of two curves/lines, it is necessary to solve the equations of the curves/lines simultaneously.

The discriminant

Whenever we use the equations of two curves/lines to generate a quadratic equation, the discriminant of this quadratic equation tells us the nature of the intersection:

• $b^2-4ac=0$ means that the two graphs intersect at exactly one point
• $b^2-4ac>0$ means that the two graphs intersect at more than one points
• $b^2-4ac<0$ means that the two graphs do not intersect

Many times we do have sufficient information to narrow down the solution to get the exact points but we can get a range of values for which the discriminant has a certain value.

Finding intersections between a curve and a line when data is missing

It is possible to get the values of an unknown $k$ given the following forms of the line:

• $y=kx+c$ (e.g. $y=kx-1$)
• $y=mx+k$ (e.g. $y=3x+k$)
• $y=kx+k$

Where $m$ and $c$ are known and we are told whether the curve and the line intersect once, multiple times or not at all.

Example