5. The Coordinate Plane & Graphing Lines

The coordinate plane is two number lines crossing at the origin \((0,0)\): a horizontal x-axis and a vertical y-axis. Every point has an address \((x, y)\): go across \(x\), then up/down \(y\).

Most GED® line questions use slope-intercept form:
\[ y = mx + b, \]
where \(m\) is the slope (steepness) and \(b\) is the y-intercept (where the line crosses the y-axis).

Slope = rise over run — how far the line goes up for each step to the right:
\[ m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}. \]

Worked example. Graph \(y = 2x + 1\).

• The y-intercept is \(b = 1\), so plot the point \((0, 1)\).
• The slope is \(m = 2 = \frac{2}{1}\): from \((0,1)\), go up 2 and right 1 to reach \((1, 3)\).
• Connect the points.

Slope from two points. Through \((1, 2)\) and \((4, 8)\): \(m = \frac{8-2}{4-1} = \frac{6}{3} = 2\).

⚠️ Common mistake: a positive slope rises left-to-right; a negative slope falls. A horizontal line has slope \(0\).

💡 Tip: to find the x-intercept, set \(y = 0\) and solve; to find the y-intercept, set \(x = 0\).