The coordinate plane and plotting points

Key idea

The coordinate plane is formed by the x-axis (horizontal) and y-axis (vertical). A point is written as an ordered pair \((x,y)\). The sign of \(x\) and \(y\) tells you the quadrant. Reflections are quick coordinate changes:

• Across the x-axis: \((x,y)\to(x,-y)\)
• Across the y-axis: \((x,y)\to(-x,y)\)

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Summary

• Coordinates are ordered pairs \((x,y)\): move \(x\) horizontally, then \(y\) vertically.
• Reflections across axes flip the sign of one coordinate.

Slope (gradient) and rate of change

Key idea

The slope \(m\) of a line through \((x_1,y_1)\) and \((x_2,y_2)\) is: \[ m=\frac{y_2-y_1}{x_2-x_1}. \] Slope measures how much \(y\) changes for each 1 unit change in \(x\) (rise over run). A horizontal line has slope \(0\). A vertical line has undefined slope because \(x_2-x_1=0\).

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Summary

• Slope formula: \(m=\dfrac{y_2-y_1}{x_2-x_1}\).
• Horizontal slope \(=0\); vertical slope is undefined.

Equation of a line: slope-intercept and point-slope

Key idea

The most common line forms are:

• Slope-intercept form: \(\;y=mx+b\) where \(m\) is slope and \(b\) is the y-intercept.
• Point-slope form: \(\;y-y_1=m(x-x_1)\) using a point \((x_1,y_1)\) and slope \(m\).

Parallel lines have the same slope. If you know the slope and one point, point-slope form is a reliable starting point.

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Summary

• Use \(y=mx+b\) when you know slope and y-intercept (or can find them).
• Use \(y-y_1=m(x-x_1)\) when you know slope and a point.
• Parallel lines share the same slope.

Intercepts and standard form \(Ax+By=C\)

Key idea

To find intercepts, use these reliable “set to zero” rules:

• y-intercept: set \(x=0\).
• x-intercept: set \(y=0\).

In slope-intercept form \(y=mx+b\), the y-intercept is \(b\). In standard form \(Ax+By=C\), it’s often fastest to substitute \(x=0\) or \(y=0\).

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Summary

• y-intercept: set \(x=0\). x-intercept: set \(y=0\).
• In \(y=mx+b\), the y-intercept is \(b\).

Parallel and perpendicular lines

Key idea

Two non-vertical lines are:

• Parallel if they have the same slope.
• Perpendicular if their slopes are negative reciprocals, so \(m_1m_2=-1\).

Special cases: a vertical line is perpendicular to a horizontal line. (Vertical slope is undefined, horizontal slope is 0.)

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Summary

• Parallel: same slope.
• Perpendicular: negative reciprocal slopes (when defined).

Distance formula and midpoint formula

Key idea

The distance between \((x_1,y_1)\) and \((x_2,y_2)\) is: \[ d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}. \] The midpoint of the segment joining the two points is: \[ M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right). \]

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Summary

• Distance uses the Pythagorean theorem idea: square, add, and square root.
• Midpoint is the average of x-values and the average of y-values.

Coordinate geometry applications and final check

Where coordinate geometry shows up

• Graphing and functions: lines, intercepts, slope as rate of change.
• Geometry and proofs: slopes and midpoints help prove shapes (like parallel sides).
• Science and data: linear models, trend lines, and interpretation of graphs.
• Navigation and design: coordinates, distances, and directions on a grid.

Worked example: special lines

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Final recap

• Coordinate plane: ordered pairs \((x,y)\), origin, axes, quadrants.
• Slope: \(m=\dfrac{y_2-y_1}{x_2-x_1}\); horizontal slope \(0\), vertical slope undefined.
• Line equations: \(y=mx+b\) and \(y-y_1=m(x-x_1)\).
• Intercepts: set \(x=0\) for y-intercept; set \(y=0\) for x-intercept.
• Parallel: same slope. Perpendicular: negative reciprocal slopes (when defined).
• Distance and midpoint: \(\sqrt{(\Delta x)^2+(\Delta y)^2}\) and \(\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\).