Question

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

Answer 1

We know that
`y = x^2` is a parabola, concave up, with vertex in the origin
`(0,0)`

So, we may use three horizontal lines for our purpose: any horizontal line above the x axis will intersect the parabola twice. The x axis itself intersects the parabola once on the vertex, while any horizontal line below the x axis won't intercept the parabola.

Here's the examples:

• The horizontal line
  `y = 4` intercepts the parabola twice: the system
  `y = x^2,\ y = 4` is solved by
  `x^2=4 \implies x = \pm 2`
• The horizontal line
  `y=0` intercepts the parabola only once: the system is
  `y=x^2,\ y=0` which yields
  `x^2=0\implies x=0` which is a repeated solution
• The horizontal line
  `y=-5` intercepts the parabola only once: the system is
  `y=x^2,\ y=-5` which yields
  `x^2=-5` which is impossible, because a squared number can't be negative.