Graphing and the Rectangular Coordinate System: Study Notes for Intermediate Algebra

Graphing and the Rectangular Coordinate System

Introduction

This section introduces the fundamental concepts of graphing in algebra, focusing on the rectangular coordinate system, plotting ordered pairs, and graphing equations. Mastery of these topics is essential for understanding functions, equations, and their graphical representations in Intermediate Algebra.

The Rectangular Coordinate System

The rectangular coordinate system (also known as the Cartesian plane) is a two-dimensional space defined by a horizontal axis (x-axis) and a vertical axis (y-axis). The intersection of these axes is called the origin (0, 0).

• Ordered pair: Two numbers (x, y) that specify the location of a point on the plane. The first number is the x-coordinate (horizontal), and the second is the y-coordinate (vertical).

• Coordinate: Each number in an ordered pair; x-coordinate and y-coordinate.

• x-axis: The horizontal number line.

• y-axis: The vertical number line.

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

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

Plotting Ordered Pairs

To plot an ordered pair (x, y):

• Start at the origin (0, 0).

• Move x units right (if x is positive) or left (if x is negative).

• Move y units up (if y is positive) or down (if y is negative).

Important: The order of coordinates matters. For example, (–4, 2) and (2, –4) are in different locations.

Each ordered pair corresponds to exactly one point in the plane, and vice versa.

Example: Plotting Points and Identifying Quadrants

• (4, 2): Quadrant I

• (–3, –2): Quadrant III

• (2, –3): Quadrant IV

• (0, 4): On the y-axis

• (5, 0): On the x-axis

Determining Solutions to Equations

An ordered pair (x, y) is a solution to an equation if substituting x and y into the equation makes it true.

• Example: Is (3, –2) a solution to $2x + 5y = –4$?

  • Substitute: $2(3) + 5(–2) = 6 + (–10) = –4$

  • Result: $–4 = –4$ (True)

  • So, (3, –2) is a solution.

• Example: Is (–1, 6) a solution to $3x – y = 5$?

  • Substitute: $3(–1) – 6 = –3 – 6 = –9$

  • Result: $–9 \neq 5$ (False)

  • So, (–1, 6) is not a solution.

Graphing Linear Equations

A linear equation in two variables can be written in standard form:

$Ax + By = C$

• A, B, and C are real numbers; A and B are not both zero.

• Other forms include slope-intercept form ($y = mx + b$) and point-slope form ($y - y_1 = m(x - x_1)$).

Steps to Graph a Linear Equation

1. Rewrite the equation in slope-intercept form if possible.

2. Choose values for x (or y) to find ordered pairs.

3. Plot the points and draw a straight line through them.

Example: Graph $y = \frac{3}{4}x + 3$ by choosing x-values that avoid fractions (multiples of 4).

Finding Intercepts

Intercepts are points where the graph crosses the axes.

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

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

Example: If a graph crosses the x-axis at (2, 0) and the y-axis at (0, –8), then 2 is the x-intercept and –8 is the y-intercept.

Example: Graphing by Intercepts

Graph $2x = y$ by finding and plotting its intercepts:

• x-intercept: y = 0, $2x = 0 \Rightarrow x = 0$

• y-intercept: x = 0, $y = 0$

Graphing Nonlinear Equations

Not all equations are linear. Nonlinear equations include parabolas and absolute value graphs.

• Parabola: $y = x^2$

• Absolute value: $y = |x|$

To graph nonlinear equations, make a table of values, plot the points, and connect them smoothly.

Example: Graphing $y = x^2$

x	y
–2	4
–1	1
0	0
1	1
2	4

Example: Graphing $y = |x|$

x	y
–2	2
–1	1
0	0
1	1
2	2

Helpful Hints

• If the equation is solved for y, choose x-values to find ordered pairs.

• If the equation is solved for x, choose y-values to find ordered pairs.

• For intercepts, remember: x-intercept (y = 0), y-intercept (x = 0).

Additional info: The included image visually reinforces the concept of graphing in the rectangular coordinate system, which is central to this chapter.