Graphs & the Rectangular Coordinate System

Introduction to the Rectangular (Cartesian) Plane

The rectangular coordinate system, also known as the Cartesian plane, is a two-dimensional plane formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). This system is fundamental for graphing equations and analyzing relationships between variables in precalculus.

• Horizontal axis: x-axis

• Vertical axis: y-axis

• Origin: The point (0, 0) where the axes intersect

• Ordered pairs (x, y): Specify the position of points on the plane

• Quadrants: The axes divide the plane into four quadrants, numbered I to IV counterclockwise starting from the upper right

• Sign conventions:

  • x > 0: right of origin; x < 0: left of origin

  • y > 0: above origin; y < 0: below origin

Example: Plotting points such as A(4, 3), B(-3, 2), C(-2, -3), D(5, -4), E(0, 0), F(0, -3) on the coordinate plane helps visualize their locations and identify their quadrants.

Solving Two Variable Equations

Equations with One vs. Two Variables

Equations can involve one or two variables. In precalculus, many equations involve both x and y, and their solutions are represented as points (x, y) on the Cartesian plane.

• To determine if a point (x, y) satisfies an equation, substitute x and y into the equation and check if the statement is true.

• The graph of an equation is the set of all points (x, y) that satisfy the equation.

• If a point satisfies the equation, it lies on the graph; otherwise, it does not.

Example: For the equation x + y = 5, check if points (3,2), (4,1), (0,5), and (-1,3) satisfy the equation and plot them on the graph.

Graphing Two Variable Equations by Plotting Points

Plotting Points to Graph Equations

To graph an equation, calculate and plot ordered pairs (x, y) that satisfy the equation. This method is especially useful for linear and simple nonlinear equations.

• Choose several x-values (often 3-5) and solve for the corresponding y-values.

• Plot each (x, y) pair on the coordinate plane.

• Connect the points with a straight line (for linear equations) or a smooth curve (for nonlinear equations).

Graphing by Plotting Points: Steps

1. Isolate y to the left side: y = ...

2. Calculate y-values for 3-5 chosen x-values

3. Plot (x, y) points from Step 2

4. Connect points with a line or curve

Example: Graph the equation -2x + y = -1 by creating ordered pairs using x = -2, -1, 0, 1, 2.

Practice: Graph y - x^2 + 3 = 0 and y = sqrt(x) + 1 by choosing points that satisfy the equations. (For the square root, choose positive x-values only.)

Graphing Intercepts

Finding x- and y-Intercepts

Intercepts are points where the graph crosses the x-axis or y-axis.

• To find the x-intercept, set y = 0 and solve for x.

• To find the y-intercept, set x = 0 and solve for y.

Example: Write the x- and y-intercepts of a given graph. For more complex graphs, identify all points where the graph crosses the axes.

Slopes of Lines

Definition and Calculation of Slope

The slope of a line measures how steep the line is. It is defined as the ratio of the change in y (vertical change) to the change in x (horizontal change) between two points on the line.

• Formula for slope (m):

• Given two points (x₁, y₁) and (x₂, y₂), substitute their coordinates into the formula.

• The order of points does not affect the slope value.

Example: Find the slopes of lines A and B shown in a graph using their respective points.

Practice: Find the slope of a line shown on a graph or the line containing two given points, such as (-1,1) and (4,3).

Summary Table: Key Concepts

Additional info:

• These notes cover foundational graphing skills essential for success in precalculus, including plotting points, graphing equations, finding intercepts, and calculating slopes.

• Mastery of these concepts is critical for understanding more advanced topics such as functions, transformations, and systems of equations.