Ordered pairs, axes, and quadrants

Key idea

The x-axis is horizontal and the y-axis is vertical. The origin is \((0,0)\). An ordered pair \((x,y)\) tells you how to move from the origin: move \(x\) units left/right, then \(y\) units down/up.

Quadrants are determined by signs: I \((+,+)\), II \((-,+)\), III \((-,-)\), IV \((+,-)\). If \(x=0\), the point is on the y-axis. If \(y=0\), the point is on the x-axis.

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Summary

• Use \((x,y)\): move \(x\) first (left/right), then \(y\) (down/up).
• Quadrants depend on the signs of \(x\) and \(y\). Points with \(x=0\) or \(y=0\) lie on an axis.

Slope: rise over run and slope between two points

Key idea

The slope \(m\) measures how \(y\) changes compared to \(x\): \[ m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}. \] A positive slope rises left to right, a negative slope falls left to right. A horizontal line has slope \(0\). A vertical line has an undefined slope because \(\Delta x=0\).

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Summary

• Slope formula: \(m=\dfrac{y_2-y_1}{x_2-x_1}\).
• Horizontal lines have slope \(0\). Vertical lines have undefined slope.

Graphing lines with slope-intercept form

Key idea

In slope-intercept form, \[ y=mx+b, \] \(m\) is the slope and \(b\) is the y-intercept (the point \((0,b)\)). To graph: (1) plot \((0,b)\), then (2) use the slope \(m=\dfrac{\text{rise}}{\text{run}}\) to find another point, then (3) draw the line through the points.

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Summary

• Slope-intercept form: \(y=mx+b\).
• Slope \(0\) makes a horizontal line \(y=c\). Vertical lines look like \(x=c\).

x-intercepts, y-intercepts, and standard form

Key idea

In standard form, \[ Ax+By=C, \] the x-intercept happens when \(y=0\), and the y-intercept happens when \(x=0\). That gives two easy points to plot.

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Summary

• x-intercept: set \(y=0\). y-intercept: set \(x=0\).
• Standard form \(Ax+By=C\) works well with the intercept method.

Writing equations of lines from slope and points

Key idea

If you know a slope \(m\) and a point \((x_1,y_1)\), the most direct form is point-slope form: \[ y-y_1=m(x-x_1). \] You can simplify to get slope-intercept form \(y=mx+b\). If a line passes through the origin \((0,0)\), then \(b=0\) and the equation is \(y=mx\).

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Summary

• Point-slope form: \(y-y_1=m(x-x_1)\).
• Convert to \(y=mx+b\) to graph quickly.

Parallel and perpendicular lines

Key idea

Parallel lines have the same slope. Perpendicular lines (forming a right angle) have slopes that are negative reciprocals: if \(m\) is the slope of one line, then the perpendicular slope is \(-\dfrac{1}{m}\) (as long as the line is not vertical/horizontal).

Two special cases: a horizontal line \(y=c\) is perpendicular to a vertical line \(x=k\).

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Summary

• Parallel lines: same slope.
• Perpendicular lines: negative reciprocal slopes (or horizontal vs vertical).

Why coordinate plane and graphing lines matter

Where lines show up

• Rate problems: slope represents speed, cost per item, or change per unit.
• Data and trends: a line models steady increase or decrease.
• Geometry on the coordinate plane: slopes help prove parallel or perpendicular relationships.
• Algebra and functions: linear functions are the foundation for graphing and solving systems.

Worked example: slope as a rate

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Final recap

• Coordinate plane: ordered pairs \((x,y)\), axes, origin, and quadrants.
• Slope: \(m=\dfrac{\Delta y}{\Delta x}\); horizontal lines have slope \(0\), vertical lines have undefined slope.
• Graphing lines: use \(y=mx+b\) (y-intercept + slope moves) or intercepts from \(Ax+By=C\).
• Equations: use point-slope form \(y-y_1=m(x-x_1)\) to build a line from a point and slope.
• Parallel: same slope. Perpendicular: negative reciprocal slope (or horizontal vs vertical).