Question

Write a linear equation that intersects y=x^2 at two points . Then write a second linear that intersects y=x^2 at just one point , and a third linear equation that does not intersect y=x^2. Explain how you found the linear equations.

Answer 1

There are many possible solutions to this problem. The simplest solutions involve horizontal lines of the form y=a for different values of a. Any such equation with a>0 will intersect y=x^2 at two points. For example, y=1 which intersects y=x^2 when x^2=1, which has solutions at x=1 and x=-1 and so the two points of intersection are (-1,1) and (1,1). A line that intersects at one point must be tangent to the parabola. The simplest equation is y=0 which intersects y=x^2 only at the point (0,0), corresponding to the fact that the equation x^2=0 has only one solution: x=0. Finally, the line y=-1 does not intersect the parabola at all since x^2=-1 has no real solutions. So possible answers for the three problems are y=1, y=0, and y=-1. (There are many other possible solutions.)