The Cartesian Plane and Ordered Pairs

Introduction to Ordered Pairs and the Cartesian Plane

The Cartesian plane is a two-dimensional coordinate system defined by a horizontal axis (x-axis) and a vertical axis (y-axis) that intersect at the origin (0,0). Each point in this plane is represented by an ordered pair (x, y), where the order of the numbers matters. This system allows us to graphically represent equations involving two variables.

• Ordered Pair: A pair of numbers (x, y) that represents a point in the plane, where x is the abscissa and y is the ordinate.

• Origin: The point (0, 0) where the x-axis and y-axis intersect.

• Quadrants: The axes divide the plane into four regions, labeled I, II, III, and IV, moving counterclockwise from the upper right.

Example: The point (–2, 3) is located 2 units left of the origin and 3 units up, placing it in Quadrant II.

Plotting Ordered Pairs and Graphing Equations

Plotting Points in the Cartesian Plane

To plot an ordered pair (x, y), move x units along the x-axis and y units along the y-axis from the origin. The intersection of these movements locates the point.

• Positive x: Move right; Negative x: Move left.

• Positive y: Move up; Negative y: Move down.

Example: To plot (–2, 3), move 2 units left and 3 units up from the origin.

Graphing Equations by Plotting Points

To graph an equation in two variables, select values for one variable, solve for the other, and plot the resulting ordered pairs. Connect the points to reveal the graph of the equation.

• Choose several x-values, solve for y, and plot the points (x, y).

• Alternatively, choose y-values and solve for x.

Example: For , if , then ; if , then ; plot (0, 3) and (1, 5).

Intercepts of a Graph

Definitions and Finding Intercepts

Intercepts are points where a graph crosses the axes.

• x-intercept: The point(s) where the graph crosses the x-axis (y = 0).

• y-intercept: The point(s) where the graph crosses the y-axis (x = 0).

Finding Intercepts Algebraically:

• To find x-intercepts: Set y = 0 and solve for x.

• To find y-intercepts: Set x = 0 and solve for y.

Example: For :

• x-intercepts: ; points: (7, 0), (–3, 0)

• y-intercept: ; point: (0, –21)

Midpoint of a Line Segment

Midpoint Formula and Applications

The midpoint of a line segment connecting two points is the point exactly halfway between them. The formula for the midpoint M between and is:

Example: Find the midpoint of A(4, –3) and B(4, 5):

Finding an Endpoint Given a Midpoint: If one endpoint and the midpoint are known, set up equations using the midpoint formula to solve for the missing endpoint.

Example: If the midpoint is (7, ½) and one endpoint is (7, 3), solve for the other endpoint (x, y):

• Other endpoint: (7, –2)

Distance Between Two Points

The Distance Formula

The distance between two points in the plane can be found using the Pythagorean Theorem. For points and , the distance d is:

• This formula is derived from the Pythagorean Theorem: .

• Always use the positive root, as distance cannot be negative.

Example: Find the distance between (–3, 2) and (7, –2):

Application: Points at a Given Distance

To find points a certain distance from a given point, use the distance formula and solve for the unknown coordinate.

Example: Find the y-coordinates of points 5 units from (–3, 3) with x = 1:

• or

• Points: (1, 6) and (1, 0)

Summary Table: Key Formulas