In this post, we will find the intersection points for three curves. The first is a curve with equation r1(t) = t and has an infinite number of intersections with both other graphs. The second is a curve with equation r2(s) = 6 − s and has four possible intersections, one at each integer value of s. Finally, the third graph is a curve that matches equation r2(s) = 6 − s but also includes the additional constraint that it must have two different values of slope at every point on its axis (i.e., it cannot be linear). We’ll use these constraints to determine when they intersect! The intersection between the curves occurs when r(t) = t, and so at any point on the x-axis where this is true. The equation for that curve (r(s) = s), would be a line with slope of – infinity (-∞). For values of t or s not on its axis, it has no intersections with either graph. As an example we can see in Figure A below: Figure B shows points along both graphs; as you’ll notice none intersect except those where their axes overlap! This makes sense since the second constraint ensures they must have distinct slopes at any given point. A similar reasoning applies to finding intersections for two more general equations, such as y=x+c