Coordinate Plane, Distances, Circles, Intercepts, and Symmetry

The plane is a 2D area formed by the x-axis (horizontal, left-right) and the y-axis (vertical, up-down).
The intersection of the axes is the origin, denoted as $0,0$.
The coordinate plane is divided into four regions called quadrants:
- Quadrant I: top-right
- Quadrant II: top-left
- Quadrant III: bottom-left
- Quadrant IV: bottom-right
Quadrants are labeled 1, 2, 3, 4 in a counterclockwise order starting from Quadrant I.
Understanding how far to go on the plane uses ordered pairs and the idea of plotting points from the origin.

Ordered pairs are written as $x, y$. They are not intervals; in this context they specify a location on the plane.
The x-coordinate tells how far to move left or right from the origin: positive x goes right, negative x goes left.
The y-coordinate tells how far to move up or down from the origin: positive y goes up, negative y goes down.
Plotting a point means moving x units along the x-axis, then y units along the y-direction from the origin.
Example points used in class (labels and coordinates):
- A = $3, 2$
- B = $-2, 4$
- C = $ -3, 0$
- D = $0, -2$
- E = $ -7, 0$ (interpreted as “negative four from negative three” along x with y = 0)
- F = $5, -1$
- G = $1, 5$
Plotting sanity check: the coordinates of G are $1, 5$, which can be read directly from the point label.
If given only a graph, you should be able to identify the coordinates of labeled points (and vice versa).

The distance between two points $x1, y1$ and $x2, y2$ is given by the distance formula:
$d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}$
Notation: use consistent labeling with $x1, y1$ and $x2, y2$.
Example: distance between G $1, 5$ and A $3, 2$:
- (x2 - x1 = 3 - 1 = 2)
- (y2 - y1 = 2 - 5 = -3)
- Distance = (\sqrt{2^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13}\)
- Approximate distance: $\sqrt{13} \approx 3.6056$
Key reminders:
- The distance is always nonnegative.
- The distance formula is symmetric: the distance from P to Q is the same as from Q to P.

The midpoint between $x1, y1$ and $x2, y2$ is:
$M = \left(\frac{x1 + x2}{2}, \, \frac{y1 + y2}{2}\right)$
Example: midpoint of G $1, 5$ and A $3, 2$:
- Midpoint = $\frac{1 + 3}{2}, \; \frac{5 + 2}{2}$ = (2, 3.5)\
Relationship to distance: the distance from either endpoint to the midpoint is half of the total distance between the endpoints.

A circle is the set of all points at a fixed distance (radius) from a fixed point (center).
Center: $h, k$ and Radius: $r$.
Standard equation of a circle:
$(x - h)^2 + (y - k)^2 = r^2$
Key circle quantities:
- Radius: the distance from center to any point on the circle, denoted $r$.
- Diameter: the distance across the circle through the center, (D = 2r\).
- Circumference: the distance around the circle,
  $C = \pi D = 2\pi r$
- Area: the area enclosed by the circle,
  $A = \pi r^2$
Interpreting the equation:
- The center is the point $h, k$.
- The right-hand side is the radius squared, (r^2\).
Important note: a circle’s edge is often described using “edge,” “circumference,” and “arc” (a piece of the circumference) or a “sector” (a portion bounded by two radii and an arc).

If you know the center $h, k$ and radius $r$, the circle is fully determined by the equation above.
Example: Circle with center $3, -2$ and radius $5$:
- Equation: $ (x - 3)^2 + (y + 2)^2 = 25 \
- Center: ( (3, -2) \
- Radius: \(r = 5$
If you know two opposite points on the circle (diametrically opposite), you can find the center as the midpoint of those two points, and the radius as the distance from the center to either point.
Example construction from two opposite edge points: suppose the opposite points are $ (-1, 4) $ and $ (-9, 4) \
- Center (midpoint): \( \left(\frac{-1 + (-9)}{2}, \frac{4 + 4}{2}\right) = (-5, 4) $
- Radius: distance from center to one endpoint, e.g. to (-1, 4): \n $r = \sqrt{(-1 - (-5))^2 + (4 - 4)^2} = \sqrt{(4)^2 + 0^2} = 4$
- Equation: \( (x + 5)^2 + (y - 4)^2 = 16 \
In general, two points on the circle determine the circle if you also know they are opposite; otherwise more information is needed.

Intercepts are points where the circle crosses the axes (the x- or y-axes).
X-intercepts: set (y = 0) and solve for (x).
Y-intercepts: set (x = 0) and solve for (y).
If a circle intersects the axes, you can have 0, 1, or 2 intercepts on each axis. The origin $0,0$ is counted as both an x-intercept and a y-intercept when it lies on the circle.
Example with circle $ (x - 3)^2 + (y + 2)^2 = 25 \
- X-intercepts (set (y = 0)): \( (x - 3)^2 + 4 = 25 \
  \Rightarrow (x - 3)^2 = 21 \
  \Rightarrow x = 3 \pm \sqrt{21} \
  \Rightarrow \left( 3 - \sqrt{21}, 0 \right), \left( 3 + \sqrt{21}, 0 \right) $
- Y-intercepts (set (x = 0)): \( (0 - 3)^2 + (y + 2)^2 = 25 \
  \Rightarrow 9 + (y + 2)^2 = 25 \
  \Rightarrow (y + 2)^2 = 16 \
  \Rightarrow y = 2 \text{ or } y = -6 \
  \Rightarrow (0, 2), (0, -6)
- Note: You may get two, one, or zero intercepts on each axis depending on the circle’s position.

Symmetry means the graph looks the same under a reflection across a line or point.
Common reflections:
- x-axis symmetry: reflect over the x-axis (y -> -y). If the equation remains unchanged when you replace y with -y, the graph has x-axis symmetry.
- y-axis symmetry: reflect over the y-axis (x -> -x). If the equation remains unchanged when you replace x with -x, the graph has y-axis symmetry.
- Origin symmetry: reflect through the origin (x -> -x, y -> -y). If the equation remains unchanged under both replacements, the graph has origin symmetry.
Quick algebraic check for a given equation F(x, y) = 0:
- If F(x, -y) = F(x, y) then the graph is symmetric about the x-axis.
- If F(-x, y) = F(x, y) then the graph is symmetric about the y-axis.
- If F(-x, -y) = F(x, y) then the graph has origin symmetry.
Example visualization:
- A circle centered at (3, 0) with radius 2 has x-axis symmetry (it looks the same above and below the x-axis) but does not have y-axis symmetry (its left and right halves are not identical).
- A circle centered at the origin has both x- and y-axis symmetry and thus origin symmetry as well.

Intercepts, symmetry, and distance/midpoint concepts tie together for solving circle-related problems:
- Intercepts help determine where a circle crosses the axes and can be used to sketch or verify a circle.
- The distance formula and midpoint concept are essential for constructing circles from opposite points on the circle and for identifying centers.
- Understanding symmetry helps in sketching and simplifying equations, and provides a quick check that a potential circle or other graph is reasonable.
Practically, circles are determined by just two pieces of information: a center $h, k$ and a radius $r$.
Real-world relevance: circle geometry is foundational in design, architecture, and any application requiring precise circular shapes and measurements.

Distance between two points:
$d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}$
Midpoint of two points:
$M = \left(\frac{x1 + x2}{2}, \, \frac{y1 + y2}{2}\right)$
Circle equation (center $h, k$, radius $r$):
$(x - h)^2 + (y - k)^2 = r^2$
Circle properties:
- Diameter: $D = 2r$
- Circumference: $C = 2\pi r$
- Area: $A = \pi r^2$

Always identify the center and radius to write the circle equation.
To find intercepts, substitute y = 0 for x-intercepts and x = 0 for y-intercepts.
Use the distance formula to compute radius when given opposite points, or use the midpoint to locate the center.
Check symmetry algebraically when possible to gain insight into the graph.