Intersection of two quadratic functions

The point where two parabolas 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 $y = -x^2 + k$.

A worked case

Setting $y = x^2$ equal to $y = -x^2 + 2$ gives $2x^2 = 2$, that is $x^2 = 1$, so $x = \pm 1$. 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 $y$, confirming that these are shared points of both.

Counting the intersections

The equation $2x^2 - k = 0$ has discriminant $D = 8k$, and its sign decides the count.

The tangent case

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

The case with no meeting

When $k = -2$, we get $x^2 = -1$, 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.

Equal leading coefficients

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.

In summary, the intersections of two quadratic functions are 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.