Basic coordinate geometry

Every point in the plane is located by an ordered pair $(x, y)$. The first number is how far the point sits horizontally from the $y$-axis, and the second is how far it sits vertically from the $x$-axis.

The figure shows why this is so useful. For point $A = (k, 7)$, drop a perpendicular to the $x$-axis at $C = (k, 0)$. The vertical leg $AC$ equals the $y$-coordinate, $7$, and the horizontal leg $BC$ from the origin $B$ equals the $x$-coordinate, $k$. The right angle sits at $C$, so a point and the origin form a right triangle whose legs you can read straight off the coordinates.

Moving points is just arithmetic on the coordinates. Shifting left by $d$ subtracts $d$ from $x$; shifting right adds $d$. Shifting down subtracts from $y$; shifting up adds. For example, shifting the point $(0, 4)$ left by $9$ gives $(-9, 4)$.

Point $A = (3, 5)$ sits in the plane, with the origin at $B = (0, 0)$. How far is $A$ from each axis? \ Its horizontal distance from the $y$-axis is the $x$-coordinate, $3$. \ Its vertical distance from the $x$-axis is the $y$-coordinate, $5$.

A circle has centre $(0, 4)$. If it is shifted $9$ units to the left, only the $x$-coordinate changes: subtract $9$ to get the new centre $(0 - 9, 4) = (-9, 4)$.