The Coordinate Plane

The coordinate plane is made from two number lines that cross at right angles. The horizontal number line is the $x$-axis. The vertical number line is the $y$-axis. The point where they cross is the origin, written $(0,0)$.

Every location on the plane is named by an ordered pair $(x,y)$. The first number tells you how far to move left or right along the $x$-axis. The second number tells you how far to move up or down parallel to the $y$-axis. Because the order matters, we call it an ordered pair: $(3,5)$ is a different point than $(5,3)$.

Distance and midpoint come from the coordinates.

The two axes split the plane into four regions called quadrants. We number them with Roman numerals, starting at the top-right and going counter-clockwise:

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• Quadrant I --- top right --- $x$ positive, $y$ positive: $(+,+)$
• Quadrant II --- top left --- $x$ negative, $y$ positive: $(-,+)$
• Quadrant III --- bottom left --- $x$ negative, $y$ negative: $(-,-)$
• Quadrant IV --- bottom right --- $x$ positive, $y$ negative: $(+,-)$

If a point sits on an axis, it is not in any quadrant.

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• A point on the $x$-axis has the form $(x,0)$ --- its $y$-coordinate is $0$. Example: $(4,0)$.
• A point on the $y$-axis has the form $(0,y)$ --- its $x$-coordinate is $0$. Example: $(0,-3)$.
• The origin $(0,0)$ sits on both axes.

To plot the point $(x,y)$:

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• Start at the origin $(0,0)$.
• Move along the $x$-axis: right if $x>0$, left if $x<0$.
• From there move parallel to the $y$-axis: up if $y>0$, down if $y<0$.
• Mark a dot where you stop.

To name the coordinates of a point you already see, do the reverse: count how far it is left/right, then up/down, from the origin.

Distance and midpoint come from the coordinates.

A table of values is a neat list of ordered pairs. We usually pick a few $x$-values, apply a rule, and record the matching $y$-values. Each row of the table $(x,y)$ becomes one point on the plane.

A rule like $y = x+2$ or $y = 2x$ describes how $y$ depends on $x$. When you plot all the ordered pairs from its table, the points fall in a perfectly straight line --- this is a linear relationship. Connecting the dots gives the graph of the rule.

Two points lie on the same horizontal line when they share the same $y$-coordinate. Two points lie on the same vertical line when they share the same $x$-coordinate. In either case the distance between them is easy to find:

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• Horizontal: subtract the $x$-coordinates and take the positive difference.
• Vertical: subtract the $y$-coordinates and take the positive difference.

Distance and midpoint come from the coordinates.

Coordinates appear everywhere: on maps, game boards, and city grids. A treasure map might place a chest at $(5,3)$, meaning $5$ blocks east and $3$ blocks north of a starting corner. Counting blocks between two spots on a grid is exactly the horizontal or vertical distance from the last section.

Distance and midpoint come from the coordinates.

For two points not on the same horizontal or vertical line, the straight-line distance still comes from a right triangle. Between $(x_1,y_1)$ and $(x_2,y_2)$, the horizontal leg has length $|x_2-x_1|$ and the vertical leg has length $|y_2-y_1|$. The distance is the hypotenuse, so by the Pythagorean theorem

Because each difference is squared, the sign of the subtraction never matters --- squaring erases it.

Distance and midpoint come from the coordinates.

The midpoint of the segment joining $(x_1,y_1)$ and $(x_2,y_2)$ is the point exactly halfway between them. You simply average the $x$-coordinates and average the $y$-coordinates:

The average of two numbers always lands in the middle, so this works even when the coordinates are negative.

The slope of a line measures how steeply it rises. It is the ratio of vertical change to horizontal change between any two points on the line:

A positive slope climbs left-to-right; a negative slope falls. A larger $|m|$ means a steeper line. In a real setting, slope is a rate of change --- for the plant rule $y=3x+1$, the slope $3$ means “$3$ cm of growth per day.”

Every non-vertical line can be written as

where $m$ is the slope and $b$ is the $y$-intercept --- the $y$-value where the line crosses the $y$-axis (that is, where $x=0$). For $y=2x+1$: the slope is $m=2$, and the line crosses the $y$-axis at $(0,1)$. Each step right of $1$ raises $y$ by $2$. This is the same “starting amount plus steady change” idea from the tables, written in one tidy formula.

Moving or flipping a shape changes each point by a simple rule:

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• Translation (slide) by $a$ right and $b$ up: $(x,y)\rightarrow(x+a,\ y+b)$.
• Reflection over the $x$-axis: $(x,y)\rightarrow(x,\ -y)$.
• Reflection over the $y$-axis: $(x,y)\rightarrow(-x,\ y)$.
• Reflection over the origin: $(x,y)\rightarrow(-x,\ -y)$.

Apply the rule to every vertex, then reconnect the new points to get the image.

You can find the area of a triangle straight from its corner coordinates $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$ --- no height needed:

The absolute-value bars keep the area positive no matter the order of the vertices. This trick is called the shoelace formula because of the criss-cross pattern of multiplications.

A lattice point is a point whose coordinates are both integers, like $(3,-2)$ --- exactly where grid lines cross. On a city grid you cannot cut diagonally through buildings, so the travel distance is the taxicab distance: add the horizontal and vertical blocks,

From $(1,2)$ to $(5,5)$ this is $|5-1|+|5-2|=4+3=7$ blocks, even though the straight-line distance is only $5$. The taxicab distance is never shorter than the straight-line distance.