Absolute Value and Coordinate Plane Study Guide

Definition and Conceptual Meaning of Absolute Value

The absolute value of a number represents its physical distance from $0$ on a standard number line. Because distance is measured as a spatial measurement, absolute value is always non-negative. Whether moving to the left or to the right of zero on a scale, the distance traveled remains positive or zero. A fundamental key phrase to internalize is: "Absolute value tells how far, not which direction!"

Mathematical Notation and Representation

Absolute value is formally written using vertical bars enclosed around a term or expression, represented as $|a|$. This mathematical notation is read aloud as "the absolute value of $a$".

Consider a horizontal number line centered at $0$. Moving $4$ units to the left along the number line locates the value $-4$. The distance from $0$ to $-4$ is $4$ units, expressed in notation as $|-4| = 4$. Similarly, moving $4$ units to the right along the number line locates the value $4$. The distance from $0$ to $4$ is also $4$ units, expressed in notation as $|4| = 4$. Both numbers $-4$ and $4$ are located exactly $4$ units away from $0$.

Core Principles of Absolute Value

There are four fundamental rules to remember regarding absolute value operations:

1. Absolute value is defined strictly as the distance of a point from $0$.

2. Distance is always non-negative (positive or zero).

3. Opposite numbers on a number line have identical absolute values (for example, $|-a| = |a|$).

4. Absolute value is used to describe the total size, scale, magnitude, or amount of a change without regard to whether the change is positive or negative.

Worked Examples and Computational Practice

Evaluating the absolute value of various individual numbers yields the following results:

\n|7| = 7\n

\n|-7| = 7\n

\n|0| = 0\n

\n|12| = 12\n

\n|-12| = 12\n

Comparing pairs of opposite numbers demonstrates that opposite signed values evaluate to identical positive magnitude outcomes:

\n|-5| = 5 \quad \text{and} \quad |5| = 5\n

\n|3| = 3 \quad \text{and} \quad |-3| = 3\n

\n|8| = 8 \quad \text{and} \quad |-8| = 8\n

Solutions to additional basic evaluation exercises are as follows:

\n|6| = 6\n

\n|-9| = 9\n

\n|15| = 15\n

\n|-20| = 20\n

\n|0| = 0\n

\n|-4| = 4\n

\n|17| = 17\n

\n|-74| = 74\n

Fundamentals of the Coordinate Plane

The Cartesian coordinate system is formed by two intersecting perpendicular number lines that divide a two-dimensional space into four distinct regions. Key terminology and structural components include:

Horizontal Axis ($x$-axis): The horizontal number line on a coordinate grid.

Vertical Axis ($y$-axis): The vertical number line on a coordinate grid.

Origin: The central point where the $x$-axis and $y$-axis intersect, defined by the coordinates $(0,0)$.

Ordered Pair: A set of two numbers $(x,y)$ that specifies the exact location of a point on the grid.

$x$-coordinate: The first value in an ordered pair $(x,y)$, which indicates horizontal positioning relative to the origin.

$y$-coordinate: The second value in an ordered pair $(x,y)$, which indicates vertical positioning relative to the origin.

Quadrant: One of the four regions divided by the axes, designated by Roman numerals I, II, III, and IV, proceeding counterclockwise starting from the top-right quadrant.

Plotting Points and Quadrant Identification

The four quadrants of the coordinate plane are categorized by the algebraic signs of their coordinates:

Quadrant I: Situated in the top-right region, where both coordinates are positive ($x > 0, y > 0$).

Quadrant II: Situated in the top-left region, where the $x$-coordinate is negative and the $y$-coordinate is positive ($x < 0, y > 0$).

Quadrant III: Situated in the bottom-left region, where both coordinates are negative ($x < 0, y < 0$).

Quadrant IV: Situated in the bottom-right region, where the $x$-coordinate is positive and the $y$-coordinate is negative ($x > 0, y < 0$).

To plot a point $A$ given by the ordered pair $(-3, 2)$, begin at the origin $(0,0)$. First, move $3$ units to the left along the horizontal $x$-axis due to the negative $x$-coordinate. From that position, move $2$ units upward parallel to the vertical $y$-axis due to the positive $y$-coordinate. Point $A$ is situated in Quadrant II.