Quadratic systems: a line and a parabola

When you solve a system of a line and a parabola, you set them equal and get a quadratic equation.
How many real solutions that quadratic has is exactly how many points the two graphs share.

Two real solutions means the line cuts the parabola at two points, and no real solution means the line misses it entirely.
The special case is exactly one solution, where the line just touches the parabola.
Such a line is called a tangent, and the quadratic has a repeated root.

A horizontal line — one parallel to the $x$-axis, like $y = c$ — can touch a parabola at only one point, and that point must be the vertex.

To find the vertex of $y = x^2 - 6x + 11$, complete the square: $x^2 - 6x + 11 = (x^2 - 6x + 9) - 9 + 11 = (x - 3)^2 + 2$.
Since $(x - 3)^2$ is never negative, the smallest $y$ can be is $2$, reached when $x = 3$, so the vertex is $(3, 2)$ and the only horizontal tangent is $y = 2$.

Sometimes a problem states outright that the single intersection is the vertex, and then you just need to find the vertex.
Suppose a vertical line $x = c$ passes through the lowest point of $y = x^2 - 10x + 30$; completing the square gives $(x^2 - 10x + 25) - 25 + 30 = (x - 5)^2 + 5$.
Because $(x - 5)^2 \ge 0$, the lowest value is $5$, reached when $x = 5$, so the lowest point is $(5, 5)$ and the line through it is $x = 5$.

How many points do $y = x^2$ and $y = 2x - 1$ share?
Set them equal: $x^2 = 2x - 1$, so $x^2 - 2x + 1 = 0$, which is $(x - 1)^2 = 0$.
There is one repeated root $x = 1$, so the line is tangent and they meet at exactly one point.

The line $y = -2$ touches the parabola $y = (x - 3)^2 - 2$ at only one point. What is the $y$-coordinate of that point?
A horizontal line touches at one point only at the vertex, and here the vertex value is $-2$.
So the $y$-coordinate of the point is $-2$.

A line meets $y = x^2 - 4x$ only at its lowest point. What is the $x$-coordinate of that point?
The lowest point is the vertex, midway between the roots $x = 0$ and $x = 4$.
So the $x$-coordinate is $\frac{0 + 4}{2} = 2$.