Intersection of two quadratic functions

The point where two parabolas (graphs of quadratic functions) meet is found the same way as before. At an intersection the two $y$ values are equal, so we set the two expressions equal to make an equation. Here we look at how the upward parabola $y = x^2$ meets the downward parabola $y = -x^2 + k$.

For example, setting $y = x^2$ equal to $y = -x^2 + 2$ gives $x^2 = -x^2 + 2$, so $2x^2 = 2$, that is $x^2 = 1$, and $x = 1, -1$. Putting these back into $y = x^2$ gives $y = 1$, so the intersection points are $(1, 1)$ and $(-1, 1)$. Either parabola gives the same $y = 1$, confirming that these are shared points of both.

Again, the number of real solutions of the equation is the number of intersection points. The equation $2x^2 = k$, that is $2x^2 - k = 0$, has discriminant $D = 8k$, and its sign decides the count: $k > 0$ gives 2 points, $k = 0$ gives one point of tangency, and $k < 0$ gives none.

When $k = 0$, that is $y = x^2$ and $y = -x^2$, we get $2x^2 = 0$, a repeated root $x = 0$. The two parabolas just touch at the origin $(0, 0)$ and then separate, one going up and the other down. This is the case where the two parabolas are tangent, and $(0, 0)$ is the point of tangency.

When $k = -2$, that is $y = x^2$ and $y = -x^2 - 2$, we get $2x^2 = -2$, so $x^2 = -1$, which has no real solution. The value of $y = x^2$ is always at least $0$, while $y = -x^2 - 2$ is always at most $-2$, so the two values can never be equal: the upward parabola stays above and the downward one below, and they never meet.

Incidentally, when the two parabolas have the same $x^2$ coefficient (for example $y = x^2$ and $y = x^2 + 3$), the $x^2$ terms cancel and the equation becomes linear, so there is at most one intersection, or none if the parabolas are simply shifted apart. It is when the $x^2$ coefficients differ that the number of intersections varies among 0, 1, and 2.

In summary, the intersections of two quadratic functions are also decided by the number of real solutions of the equation you get by setting the expressions equal. The large dots on the graph mark the points of intersection and tangency.