Distance and Midpoint Formulas; Graphs, Intercepts, and Symmetry in the Coordinate Plane

2.1 The Distance and Midpoint Formulas

The Coordinate Plane

The coordinate plane consists of two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). Their intersection point is called the origin, denoted as (0, 0). Any point in the plane is represented by an ordered pair (x, y), where x is the horizontal coordinate and y is the vertical coordinate.

• Quadrants: The plane is divided into four quadrants:

  • Quadrant I: (+, +) — both x and y are positive (e.g., (1, 2))

  • Quadrant II: (−, +) — x is negative, y is positive (e.g., (−3, 1))

  • Quadrant III: (−, −) — both x and y are negative (e.g., (−4, −1))

  • Quadrant IV: (+, −) — x is positive, y is negative (e.g., (3, −2))

Pythagorean Theorem Refresher

The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides:

$a^2 + b^2 = c^2$

Distance Formula

The distance between two points $P_1 = (x_1, y_1)$ and $P_2 = (x_2, y_2)$ in the plane is given by:

$d(P_1, P_2) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Example: Find the distance between $P_1 = (1, 2)$ and $P_2 = (4, 3)$.

  • $d = \sqrt{(4-1)^2 + (3-2)^2} = \sqrt{9 + 1} = \sqrt{10} \approx 3.16$

Midpoint Formula

The midpoint $M$ of the segment joining $P_1 = (x_1, y_1)$ and $P_2 = (x_2, y_2)$ is:

$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

• Example: Find the midpoint between $P_1 = (5, 5)$ and $P_2 = (3, 1)$.

  • $M = \left( \frac{5+3}{2}, \frac{5+1}{2} \right) = (4, 3)$

2.2 Graphs of Equations in Two Variables; Intercepts; Symmetry

Equations in Two Variables

An equation in two variables (x and y) is a statement where two expressions involving x and y are equal. The graph of such an equation is the set of all points (x, y) that satisfy the equation.

• Example: Does (2, 3) satisfy $x^2 + y = 7$? Substitute: $2^2 + 3 = 4 + 3 = 7$ (True).

Graphing Equations

1. Make a table of x-values (inputs) and compute corresponding y-values (outputs).

2. Plot the resulting (x, y) points on the coordinate plane.

Intercepts

Intercepts are points where a graph crosses or touches the axes.

• x-intercept: Set y = 0 and solve for x.

• y-intercept: Set x = 0 and solve for y.

• There can be multiple x-intercepts but only one y-intercept for a function.

• Example: For $y = x^2 - 2x - 3$:

  • x-intercepts: $x^2 - 2x - 3 = 0 \implies (x-3)(x+1) = 0 \implies x = 3, -1$

  • y-intercept: $y = (0)^2 - 2(0) - 3 = -3$

  • Intercepts: (3, 0), (−1, 0), (0, −3)

Symmetry of Graphs

A graph may be symmetric with respect to the x-axis, y-axis, or the origin:

• x-axis symmetry: For every (x, y), (x, −y) is also on the graph.

• y-axis symmetry: For every (x, y), (−x, y) is also on the graph.

• Origin symmetry: For every (x, y), (−x, −y) is also on the graph.

Visualizing Symmetry

• x-axis symmetry: Reflects across the x-axis.

• y-axis symmetry: Reflects across the y-axis.

• Origin symmetry: Rotational symmetry about the origin (180°).

Testing for Symmetry

Type of Symmetry	Test
x-axis	Replace y with −y in the equation. If the equation is unchanged, the graph is symmetric about the x-axis.
y-axis	Replace x with −x in the equation. If the equation is unchanged, the graph is symmetric about the y-axis.
Origin	Replace x with −x and y with −y. If the equation is unchanged, the graph is symmetric about the origin.

• Example: Test $x^2 = 2$ for symmetry:

  • x-axis: Replace y with −y (no y present, so symmetry holds).

  • y-axis: Replace x with −x: $(−x)^2 = x^2 = 2$ (symmetry holds).

  • Origin: Replace x with −x and y with −y: $(−x)^2 = x^2 = 2$ (symmetry holds).

Additional info: Symmetry helps in sketching graphs efficiently and understanding the nature of equations.