Coordinates, tables and plotting points

An ordered pair has two distinct positions

An ordered pair records two values in a stated order. In the coordinate convention used here, (x, y) gives the horizontal coordinate first and the vertical coordinate second. The point (2, 3) has horizontal coordinate two and vertical coordinate three. The point (3, 2) swaps those roles and is a different point.

Name the axes before interpreting the values. A scalar plot can label them x and y. A context plot might use elapsed time t in hours horizontally and distance d in kilometres vertically; its pair order is then (t, d). A pair does not explain its own units or meaning. Carry the axis names and units with it.

Locate zero and read each signed direction

In this Cartesian convention, the horizontal axis increases to the right and the vertical axis increases upward. Their intersection is the origin, (0, 0). Positive x is to its right and negative x to its left; positive y is above it and negative y below it. The signs describe positions relative to the axes, not a point’s size or importance.

For (−2, 3), locate −2 on the horizontal axis and three on the vertical axis. The point is two coordinate units left of the vertical axis and three above the horizontal axis. For (2, −3), it is two units right and three below. Read each sign separately rather than making both coordinates negative whenever one is negative.

Plot two coordinates at one intersection

To plot (2, 3), locate two on the horizontal axis, then locate the level three units above the horizontal axis. Their corresponding vertical and horizontal guide lines meet at the point. The guides describe its coordinates; they do not imply that the point physically travels along a path.

To read a plotted point, reverse the construction. Project vertically to the horizontal axis to read the first coordinate and horizontally to the vertical axis to read the second. Optional projection guides make that correspondence visible. Check the tick values and write the pair in the named order; a clear location with reversed coordinates still gives the wrong answer.

Use signs to identify quadrants, and keep axes separate

The two axes divide the plane into four open quadrants. Quadrant I is upper right, with x > 0 and y > 0; II is upper left, with x < 0 and y > 0; III is lower left, with both coordinates negative; IV is lower right, with x > 0 and y < 0. The Roman numerals run counterclockwise from the upper right.

A point on an axis does not belong to any quadrant. In (0, 3), the horizontal coordinate is zero, so the point lies on the vertical axis. In (2, 0), the vertical coordinate is zero, so it lies on the horizontal axis. The origin lies on both axes. Do not place an axis point into a quadrant by looking only at its nonzero coordinate.

Read tick values instead of merely counting squares

An axis scale tells you which numerical values correspond to its positions. Equal intervals on a linear axis represent equal changes in that axis’s quantity. A graph may use different intervals or units on its two axes. If horizontal ticks are 0, 2, 4 and vertical ticks are 0, 5, 10, a point one tick interval right and one interval up from the origin is (2, 5), not (1, 1).

For a time–distance plot, label the horizontal axis t (h) and the vertical axis d (km). Equal numerical steps of one then mean one hour on one axis and one kilometre on the other. They are different dimensions; equal screen spacing does not turn an hour into a kilometre. Always identify the quantity, unit, increasing direction and tick interval before reading a point.

Locate fractions between labelled values

Coordinates are not limited to whole numbers or visible grid intersections. The scalar point (1/2, −3/2) is half a coordinate unit right of the vertical axis and one and a half units below the horizontal axis. Its first coordinate lies halfway between zero and one; its second lies halfway between −1 and −2.

Keep the exact fraction when it is supplied. On an axis whose tick interval is two units, a change of 1/2 unit is one quarter of a tick interval, not half an interval. An approximate visual reading of a small display is different from an exact pair stated in the table. The explorer uses exact supplied coordinates and places the corresponding markers without rounding their values.

Transfer each table row as one ordered pair

A two-column table can record the two coordinates of each point. Read both entries from the same row, in the axis order. In a scalar table with columns x and y, the row x = −1, y = −1 gives (−1, −1). Taking x from one row and y from another creates a new pair that the table may never have supplied.

For a formula y = 2x + 1, first choose an allowed x, evaluate y, and record the pair together. At x = 1/2, y = 2 × (1/2) + 1 = 2, so the table row and plotted point are (1/2, 2). Column headings preserve the same roles as axis labels. If quantities have units, put those units in the headings and retain them when explaining a row.

Separate supplied samples from a rule that generates them

A finite table records its supplied pairs. By itself it does not establish one unique relationship at every other input. If observed whole-number inputs zero, one and two have outputs one, three and five, adding two to each output would predict seven next. Repeating the output block one, three, five would instead predict one next. Both descriptions fit those three rows.

A stipulated rule is additional information. When the question states y = 2x + 1 with real inputs allowed, you can compute further pairs from that rule and domain. At x = 2, it gives y = 5. That is a permitted pair even though it falls outside the explorer’s vertical viewing interval from −4 to 4. The display window is not the rule’s domain, and five plotted samples are not the whole graph.

Carry context and units into every pair

The explorer’s context table stipulates travel at constant speed 2 km/h, starting with distance zero at t = 0. It uses d = (2 km/h)t for elapsed times t ≥ 0. At t = 1/2 h, d = 1 km; the pair means (1/2 h, 1 km). The first entry is time and the second is distance, not two interchangeable lengths.

This stated model also gives d = 3/2 km at t = 3/4 h, even though that pair is not one of the four selected table rows. Negative elapsed time is excluded from the model’s domain, despite negative numerical positions being visible in the fixed coordinate frame. Real journey observations would need their own conditions and measurement information; these samples illustrate the stipulated exact model.

Draw a connection only when its meaning is justified

Separate sample points show the selected pairs. Joining them with segments introduces a statement about positions between those pairs. It might represent a stated continuous model, an interpolation convention or an actual connected boundary, but none of those meanings follows automatically from a finite list of dots. A list of unrelated locations is not a path merely because its rows are adjacent.

For the explorer’s stated formula and constant-speed model, the rule lets you calculate additional permitted pairs. The explorer still shows individual samples rather than drawing the full relationship. Its optional projection guides connect a selected marker to the axes so you can read its coordinates; they do not join samples to one another or assert an intermediate-value rule.

Read the same pair in the table and plot

Start with the free scalar points and selected pair (2, 3), with projection guides on. The table highlights that pair and the plot emphasizes its marker. Compare (3, 2), then inspect signed, axis, origin and fractional examples. This free collection supplies nine separate points and does not claim one formula generating them.

The rule table supplies five selected pairs from y = 2x + 1; the context table supplies four selected pairs from the stated 2 km/h model. Together there are 9 + 5 + 4 = 18 selectable samples. Turning projection guides on or off gives 18 × 2 = 36 states. All samples in the chosen dataset remain individual plotted points, with the selected one emphasized; no sample-to-sample lines are drawn.

Both numerical axes retain the fixed viewing interval −4 to 4 and equal numerical scale steps across datasets. Read x and y as scalar coordinates in the first two datasets, and t (h) and d (km) in the context dataset. Switching datasets resets selection to that dataset’s first pair and retains the guide choice. Changing the selected pair or guides does not alter the dataset’s supplied values or domain.

Check that words, table and plot describe the same values

A description, table row and plotted marker can represent the same point. Check their first and second coordinate roles, signs, exact values, units and axis scales. A point’s quadrant depends on the signs in the stated coordinate convention; axis points and the origin remain separate cases. Read the labels before interpreting the picture.

Keep the claim’s scope clear: a selected pair, an observed collection and a stipulated relationship are different kinds of information. Use the stated rule and domain to justify additional values or connections. The next lesson, Straight lines, slope and intercepts, develops how a line expresses a relationship and how its changes and axis crossings can be interpreted.

Worked examples

Try it yourself

1. What do (2, 3) and (3, 2) mean on scalar horizontal x and vertical y axes? Are they the same point? If a context instead labels horizontal time and vertical distance, which quantity goes first?

   Show the answer and reasoning

   (2, 3) means x = 2, y = 3, while (3, 2) means x = 3, y = 2; they differ. With time horizontal and distance vertical, time goes first and distance second.

   An ordered pair preserves the named coordinate roles. Axis labels and units supply its meaning; swapping entries generally changes the represented point.

2. Classify (−2, 3), (−2, −3), (2, −3), (0, 3), (2, 0) and (0, 0) as quadrant or axis locations. Why is (0, 3) not a quadrant I point?

   Show the answer and reasoning

   The first three are in quadrants II, III and IV. (0, 3) is on the vertical axis, (2, 0) on the horizontal axis, and (0, 0) is the origin on both. A zero horizontal coordinate puts (0, 3) on a boundary axis, outside the quadrants.

   Quadrants require both coordinates to be nonzero with the specified signs. The axes and their intersection are distinct location cases.

3. Horizontal ticks read 0, 2, 4 and vertical ticks read 0, 5, 10. A marker is one tick interval right and one up from the origin. Write its pair. Why does counting one square in each direction not give (1, 1)?

   Show the answer and reasoning

   The pair is (2, 5). One horizontal interval represents two units, while one vertical interval represents five.

   Position must be translated through each axis’s tick values. Equal screen intervals need not represent equal numerical changes or the same physical quantity.

4. Describe (1/2, −3/2) exactly and name its quadrant. With two-unit horizontal tick intervals, is the horizontal coordinate halfway through the interval from zero to two?

   Show the answer and reasoning

   It is half a unit right and three halves of a unit below, in quadrant IV. On a two-unit horizontal interval, 1/2 occupies one quarter, not one half.

   The coordinate fraction and the fraction of a display interval are different quantities. Read the numerical scale and preserve the supplied exact value.

5. For the stated rule y = 2x + 1 with real x allowed, find the pairs at x = −1 and x = 1/2. What is the pair at x = 2, and is it forbidden because y is outside the explorer’s −4 to 4 viewing interval?

   Show the answer and reasoning

   The pairs are (−1, −1), (1/2, 2) and (2, 5). The last is allowed by the rule and domain; its vertical value is simply outside the viewing window.

   Evaluate the rule at each supplied input and keep both entries from the same case together. A finite display range does not define the mathematical domain.

6. In the stated 2 km/h model with elapsed time t ≥ 0, write the pair at t = 2 h for horizontal t (h) and vertical d (km). Are the entries interchangeable? Does a visible negative horizontal position permit negative elapsed time?

   Show the answer and reasoning

   The pair is (2 h, 4 km). The time and distance entries are not interchangeable, and negative elapsed time remains excluded even when negative numerical positions are visible.

   The axis order, units and contextual domain remain fixed. Screen geometry alone cannot supply permission for a physically excluded input.

7. An observed table gives (0, 1), (1, 3) and (2, 5), without a stated rule. Must the next whole-number input have output seven? Must you join the points? How does a question that explicitly stipulates y = 2x + 1 differ?

   Show the answer and reasoning

   Seven is a possible continuation, but the observations alone do not determine it uniquely or require joined points. A stated rule y = 2x + 1 gives seven at x = 3 and determines other allowed pairs through the rule and its domain.

   A finite collection and a stipulated relationship make different claims. Joining samples needs an explained model or convention, and projection guides merely help read one point’s coordinates.

What to carry forward

• An ordered pair uses stated roles: in this convention the horizontal coordinate comes first and the vertical coordinate second.
• Use signs relative to the origin; quadrants exclude axis points and the origin.
• Read axis names, units and tick scales before translating positions into values, and retain exact fractions when supplied.
• Each table row gives one paired case; words, table and plot should preserve the same coordinate order and meaning.
• Selected samples do not determine a unique full relationship or automatically justify connecting lines; use a stated rule, domain and explained convention.