Graphs, equations and meeting points

The graph of $y = x^2 - 3x + 1$ is every point $(x, y)$ that makes the equation true. Nothing more, nothing less.

So "is this point on the curve?" means "do its coordinates fit the equation?"

Is $(2, -1)$ on it? Put $x = 2$ into the right-hand side:

$2^2 - 3(2) + 1 = 4 - 6 + 1 = -1$

That matches $y = -1$ ✓. Yes, it's on the curve.

Is $(1, 1)$ on it? Put $x = 1$ in:

$1 - 3 + 1 = -1$

But the point has $y = 1$, and $-1 \neq 1$. No, it's not on the curve.

Where $y = f(x)$ and $y = g(x)$ meet, both equations are true at once, so $f(x) = g(x)$ there.

• The $x$-coordinates of the meeting points are the solutions of $f(x) = g(x)$.
• The number of meeting points is the number of real solutions.

Special case: the $x$-axis is $y = 0$, so where $y = f(x)$ crosses the $x$-axis gives the roots of $f(x) = 0$.