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:

Distance Formula

The distance between two points and in the plane is given by:

• Example: Find the distance between and .

Midpoint Formula

The midpoint of the segment joining and is:

• Example: Find the midpoint between and .

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 ? Substitute: (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 :

  • x-intercepts:

  • y-intercept:

  • 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

• Example: Test for symmetry:

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

  • y-axis: Replace x with −x: (symmetry holds).

  • Origin: Replace x with −x and y with −y: (symmetry holds).

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