The Coordinate Plane and Graphing

Four Quadrants

The coordinate plane is divided into four quadrants:

| Quadrant | $x$ | $y$ | Example | |----------|-----|-----|---------| | I | + | + | $(3, 2)$ | | II | – | + | $(-3, 2)$ | | III | – | – | $(-3, -2)$ | | IV | + | – | $(3, -2)$ |

Graphing Linear Equations

A linear equation graphs as a straight line.

To graph $y = 2x - 1$:

| $x$ | $y = 2x - 1$ | Point | |-----|-------------|-------| | -1 | $2(-1) - 1 = -3$ | $(-1, -3)$ | | 0 | $2(0) - 1 = -1$ | $(0, -1)$ | | 1 | $2(1) - 1 = 1$ | $(1, 1)$ | | 2 | $2(2) - 1 = 3$ | $(2, 3)$ |

Plot the points and connect them with a straight line.

X-intercept and Y-intercept

• Y-intercept: Where the line crosses the y-axis ($x = 0$)
• X-intercept: Where the line crosses the x-axis ($y = 0$)

For $y = 2x - 4$:

• Y-intercept: $(0, -4)$
• X-intercept: Set $y = 0$: $0 = 2x - 4 \implies x = 2$, so $(2, 0)$

Horizontal and Vertical Lines

• Horizontal: $y = c$ (example: $y = 3$)
• Vertical: $x = c$ (example: $x = -2$)

Distance on the Coordinate Plane

For horizontal/vertical distances, subtract coordinates:

• Distance from $(1, 3)$ to $(5, 3)$: $|5 - 1| = 4$ units

Practice: Graph $y = -x + 3$ by making a table of values. Find the x-intercept and y-intercept.