Graphing a Table and Reading Points in Context | Grade 7 Math | LevelKite

You already know how to plot a point. Now you will use that skill on real data. A table of values lists numbers in pairs that belong together: each row matches one value of a first quantity (something you count or measure) with one value of a second quantity. A graph shows those same pairs as points on the coordinate plane, the grid formed by a horizontal axis and a vertical axis that cross at the origin, the point $(0, 0)$ . The table gives the exact numbers; the graph lets you see them all at once.

In this lesson you turn each row of a table into a point, and then you explain what a point means in a real situation. Every value in this lesson is 0 or greater, so every point lies in the first quadrant (the region to the right of the vertical axis and above the horizontal axis) or on one of the two axes along its edge.

Start with a seedling that you measure once a week. Let $x$ be the number of weeks since planting, and let $y$ be the seedling's height in centimeters.

A seedling's height, measured once a week

Weeks since planting (x)	Height in centimeters (y)
$1$	$2$
$2$	$3$
$3$	$5$
$4$	$6$

Look at one row: 3 weeks, 5 centimeters. These two numbers belong together, so you keep them together as an ordered pair: two numbers written in a fixed order inside parentheses, here $(3, 5)$ . The first number is the $x$ -coordinate and the second number is the $y$ -coordinate. In this table the $x$ column holds weeks and the $y$ column holds centimeters, so $(3, 5)$ means 3 weeks since planting, 5 centimeters tall.

On a graph of this table, the horizontal axis measures the $x$ quantity and the vertical axis measures the $y$ quantity: weeks are counted across and centimeters are counted up. Writing a whole row as one pair keeps each height matched with its own week when the numbers move from the table to the graph.

Key concept

One row, one point, one fact

Each row of a two-column table of values is one ordered pair $(x, y)$ , and each ordered pair is one point.

To plot when no value is negative: start at the origin $(0, 0)$ , move across to the right by $x$ , then move up by $y$ . A 0 means no move in that direction. Every point plotted this way lies in the first quadrant or on an axis at its edge.

To interpret a labeled point: take its exact coordinates from the label. Then say the $x$ -coordinate with the quantity and unit of the horizontal axis, and the $y$ -coordinate with the quantity and unit of the vertical axis. One point states one fact about the situation.

Worked example

Graph a table of values

The seedling table pairs $x$ weeks since planting with a height of $y$ centimeters. Its rows are 1 week and 2 centimeters, 2 weeks and 3 centimeters, 3 weeks and 5 centimeters, and 4 weeks and 6 centimeters. Graph the table.

Decide what each axis measures. The x column is weeks since planting, so weeks are counted across on the horizontal axis. The y column is height in centimeters, so centimeters are counted up on the vertical axis.
Write each row as an ordered pair with the x value first. Putting both numbers of a row into one pair keeps each height matched with its own week.
$(1, 2),\quad (2, 3),\quad (3, 5),\quad (4, 6)$
Check the condition for the plotting rule: every x value and every y value must be 0 or greater. All eight numbers in this table are greater than 0, so every move goes to the right and then up, and every point lands in the first quadrant.
$x \ge 0 \quad \text{and} \quad y \ge 0$
Plot the first pair, (1, 2). Start at the origin (0, 0), move 1 unit across for 1 week, then move 2 units up for 2 centimeters. Because you start from 0 in both directions, the moves equal the coordinates. Mark the point and label it (1, 2).
$(0 + 1,\ 0 + 2) = (1, 2)$
Plot the other three pairs with the same rule, which still applies because none of their values is negative. Start again from the origin each time: 2 across and 3 up, 3 across and 5 up, 4 across and 6 up. Label each point with its pair.
$(2, 3),\quad (3, 5),\quad (4, 6)$
Check the graph against the table. The table has four different rows, so the graph has four points, and each label repeats one row. The table records nothing between the weekly measurements, so nothing is drawn between the points.

Answer

The graph of the table is the four points $(1, 2)$ , $(2, 3)$ , $(3, 5)$ and $(4, 6)$ , shown in the figure below with weeks counted across and centimeters counted up.

The horizontal axis counts weeks since planting and the vertical axis counts height in centimeters. Each label is one row of the seedling table.

Now go the other way, from a point back to the situation. A point alone is only a position. It becomes a fact once you know what each axis measures and in which unit: the $x$ -coordinate is an amount of the horizontal-axis quantity, and the $y$ -coordinate is the matching amount of the vertical-axis quantity. In the seedling graph above, the point labeled $(4, 6)$ says that 4 weeks after planting, the seedling is 6 centimeters tall.

The next figure comes from an arcade visit where each game costs 1 token. As with every figure in this lesson, the text beside it tells you what the axes measure. The horizontal axis shows the number of games played, counted in games. The vertical axis shows the number of tokens left, counted in tokens. The five points record the visit after 0, 2, 4, 6 and 8 games. Every point is labeled with its exact coordinates, so you read the numbers from the label.

The horizontal axis counts games played and the vertical axis counts tokens left. The label on each point gives its exact coordinates.

Worked example

Say what each point means

In the arcade figure above, the horizontal axis shows the number of games played and the vertical axis shows the number of tokens left. Each game costs 1 token. What do the points labeled $(0, 8)$ , $(6, 2)$ and $(8, 0)$ tell you? Do the first two of these points agree with the price of 1 token per game?

Attach a quantity and a unit to each coordinate. No value here is negative, and each point was placed from the origin by moving across for games played and then up for tokens left. So the first number in a label counts games and the second number counts tokens.
$(x,\ y) = (\text{games played},\ \text{tokens left})$
Read the label (0, 8). The x-coordinate is 0, so no games have been played yet. The y-coordinate is 8, so 8 tokens are left. The point sits on the vertical axis because the move across is 0.
$x = 0 \text{ games}, \quad y = 8 \text{ tokens}$
Read the label (6, 2). By the matching in the first step, the x-coordinate 6 is a number of games and the y-coordinate 2 is a number of tokens, so after 6 games, 2 tokens are left.
$x = 6 \text{ games}, \quad y = 2 \text{ tokens}$
Read the label (8, 0). The x-coordinate is 8 games and the y-coordinate is 0 tokens, so after 8 games no tokens are left. The point sits on the horizontal axis because the move up is 0.
$x = 8 \text{ games}, \quad y = 0 \text{ tokens}$
Compare (0, 8) with (6, 2). Subtract the x-coordinates to count the games played between the two points, and subtract the y-coordinates to count the tokens used. Six tokens for 6 games is 1 token for each game, so the points agree with the price.
$6 - 0 = 6 \text{ games}, \quad 8 - 2 = 6 \text{ tokens}$

Answer

The point $(0, 8)$ shows 8 tokens before any game is played, $(6, 2)$ shows 2 tokens left after 6 games, and $(8, 0)$ shows no tokens left after 8 games. From $(0, 8)$ to $(6, 2)$ , 6 games used 6 tokens, so the points agree with 1 token per game.

A trail map on which every 1 centimeter stands for 2 kilometers of real trail

Length on the map in centimeters (x)	Real distance in kilometers (y)
$0$	$0$
$1$	$2$
$2$	$4$
$3$	$6$
$4$	$8$

Worked example

Graph a table, then interpret its points

On a trail map, every 1 centimeter stands for 2 kilometers of real trail. In the table above, $x$ is a length on the map in centimeters and $y$ is the real distance it stands for in kilometers. Graph the table. Then explain what the points $(0, 0)$ and $(1, 2)$ mean, and use the graph to find the map length of a trail that is really 6 kilometers long.

Name the axes and write the pairs. Map length x, in centimeters, is counted across on the horizontal axis. Real distance y, in kilometers, is counted up on the vertical axis. No value is negative, so the across-then-up rule works for every row.
$(0, 0),\quad (1, 2),\quad (2, 4),\quad (3, 6),\quad (4, 8)$
Plot each pair from the origin: across by the x-coordinate, then up by the y-coordinate. For (0, 0) both moves are 0, so the point is the origin itself. For (4, 8) you move 4 across and 8 up. Label every point with its pair.
$(0 + 4,\ 0 + 8) = (4, 8)$
Interpret (0, 0). The x-coordinate is 0 centimeters on the map and the y-coordinate is 0 kilometers of trail. No length on the map stands for no real distance.
$x = 0 \text{ centimeters}, \quad y = 0 \text{ kilometers}$
Interpret (1, 2). A length of 1 centimeter on the map stands for 2 kilometers of trail. This point states the map's scale: the real distance that a single centimeter stands for.
$x = 1 \text{ centimeter}, \quad y = 2 \text{ kilometers}$
Find the 6-kilometer trail. Real distance is the y-coordinate, so look for the label whose second number is 6. That label is (3, 6), and its first number, 3, is the map length in centimeters. Check with the map's rule: 3 centimeters at 2 kilometers each.
$3 \times 2 = 6$

Answer

The graph is the points $(0, 0)$ , $(1, 2)$ , $(2, 4)$ , $(3, 6)$ and $(4, 8)$ in the figure below. The point $(0, 0)$ means no map length and no real distance, $(1, 2)$ means 1 centimeter stands for 2 kilometers, and $(3, 6)$ shows that a 6-kilometer trail is 3 centimeters long on the map.

The horizontal axis counts length on the map in centimeters and the vertical axis counts real distance in kilometers. Each label is one row of the trail-map table.

A table and a graph are two views of the same pairs. Each row of a two-column table is one ordered pair $(x, y)$ . When no value is negative, you plot the pair from the origin $(0, 0)$ : across by $x$ , then up by $y$ . To interpret a labeled point, take its exact coordinates from the label and say each one with the quantity and unit of its axis. Keep the order, and each point you plot or read states one true fact about the situation.