Chapter 12 · The coordinate plane

Graph Real-World Data and Tell What Each Point Means

Turn a table of real-world data into points on a graph. Learn what each coordinate means in the situation and use the graph to answer questions.

14 min read

You already know how to plot, or mark, the point for an ordered pair. An ordered pair is two numbers (x,y)(x, y) that name one point on the coordinate plane, the grid made by two number lines called axes. In this lesson the numbers come from real life, so each number means something.

Ana planted a bean. Each day after it sprouted, she measured how tall the sprout was. Her measurements are data: facts collected by counting or measuring.

Ana's data has two quantities. A quantity is something you can count or measure. Her quantities are the time in days and the height in centimeters. Each row of her table pairs one time with one height.

You will graph a table like Ana's. To graph data means to show it as points on the coordinate plane. Then you will say what each point means and use the graph to answer questions.

Ana's bean sprout
Days after sproutingHeight in centimeters
11
22
22
33
33
55
44
88

A graph can show Ana's whole table in one picture. Each row becomes one point. Before you plot, decide what each axis stands for.

The x-axis is the number line that runs left to right. The y-axis is the number line that runs from bottom to top. Each axis stands for one quantity and its unit. A unit is what you count or measure in, such as days or centimeters. Ana has two quantities, so each one gets its own axis.

When one quantity is time, a common choice is to put time on the x-axis. Then the height goes on the y-axis. With this choice, a point farther right shows a later day, and a point higher up shows a taller sprout. You could choose the other way around. What matters is that you say your choice and keep it for every point.

Your choice sets the order of the numbers. In an ordered pair (x,y)(x, y), the first number is the x-coordinate and goes with the x-axis. The second number is the y-coordinate and goes with the y-axis. So with days on the x-axis and height on the y-axis, you write every pair as (days, height).

Worked example

Graph Ana's table

Graph the data in Ana's table. Say what each axis stands for.

  1. Decide what each axis stands for. One quantity is time, so let the x-axis stand for days after sprouting. Let the y-axis stand for height in centimeters. Now every ordered pair has days first and height second.

    (x, y)=(days, centimeters)(x,\ y) = (\text{days},\ \text{centimeters})

  2. Write each row of the table as an ordered pair, with days first because days are on the x-axis. The first row has 1 day and 2 centimeters, so it becomes the pair with x-coordinate 1 and y-coordinate 2. Do the same for the other three rows.

    (1,2), (2,3), (3,5), (4,8)(1, 2),\ (2, 3),\ (3, 5),\ (4, 8)

  3. Plot each pair the way you already know. For (3, 5), start at the origin, the point where the two axes meet. Move 3 to the right for 3 days. Then move 5 up for 5 centimeters. Plot the other three pairs the same way.

    (3,5): 3 right, 5 up(3, 5):\ 3 \text{ right},\ 5 \text{ up}

Answer

Ana's graph has four points: (1,2)(1, 2), (2,3)(2, 3), (3,5)(3, 5) and (4,8)(4, 8). The x-axis stands for days after sprouting, and the y-axis stands for height in centimeters. You can see the points in the graph below.

Coordinate grid from x 0 to 6 and y 0 to 10, with a grid step of 1, showing 4 plotted elements. Details: point at (1, 2), named (1, 2); point at (2, 3), named (2, 3); point at (3, 5), named (3, 5); point at (4, 8), named (4, 8).
Ana's graph: the x-axis stands for days and the y-axis stands for centimeters. The point (3,5)(3, 5) is 3 to the right for 3 days and 5 up for 5 centimeters.

Key concept

Each point tells two facts that go together

When you graph a table of real-world data, each point is one row of the table. Its x-coordinate is the value, or amount, of the quantity on the x-axis. Its y-coordinate is the value of the quantity on the y-axis. The two values go together because they come from the same row.

On Ana's graph, the x-axis stands for days and the y-axis stands for centimeters. So the point (4,8)(4, 8) means: 4 days after sprouting, the sprout was 8 centimeters tall.

To say what a point on a graph like this means, say each coordinate with its axis quantity and its unit.

Coordinate grid from x 0 to 6 and y 0 to 10, with a grid step of 1, showing 5 plotted elements. Details: point at (0, 3), named (0, 3); point at (1, 4), named (1, 4); point at (2, 6), named (2, 6); point at (3, 7), named (3, 7); point at (4, 9), named (4, 9).
Leo's graph: the x-axis stands for weeks since he started saving and the y-axis stands for dollars in his jar. The point (3,7)(3, 7) is 3 to the right and 7 up.

Worked example

Say what a point means

Leo saves money in a jar. He counted the money when he started and at the end of each of the next 4 weeks. You can see his graph above. The x-axis stands for weeks since he started saving. The y-axis stands for dollars in the jar. What do the points (3,7)(3, 7) and (0,3)(0, 3) mean?

  1. Start with the point (3, 7). Its x-coordinate is 3. The x-axis stands for weeks, so the 3 means 3 weeks since Leo started saving.

    x-coordinate 3 means 3 weeksx\text{-coordinate } 3 \text{ means } 3 \text{ weeks}

  2. Its y-coordinate is 7. The y-axis stands for dollars, so the 7 means 7 dollars in the jar. Both numbers belong to the same point, so put them in one sentence: after 3 weeks, the jar held 7 dollars.

    y-coordinate 7 means 7 dollarsy\text{-coordinate } 7 \text{ means } 7 \text{ dollars}

  3. Read the point (0, 3) the same way. Its x-coordinate is 0, so 0 weeks have passed. That is the moment Leo started. Its y-coordinate is 3, so the jar held 3 dollars. The point sits on the y-axis because it is 0 units to the right.

    (0,3) means 0 weeks, 3 dollars(0, 3) \text{ means } 0 \text{ weeks},\ 3 \text{ dollars}

Answer

The point (3,7)(3, 7) means that 3 weeks after Leo started saving, his jar held 7 dollars. The point (0,3)(0, 3) means that when he started, the jar already held 3 dollars.

A graph can also answer questions about the situation. Each point ties one value on the x-axis to one value on the y-axis. So when a plotted point has the value you know, that point shows you the value that goes with it.

  • You know the value on the x-axis, such as a time. Find it on the x-axis. Go straight up to the point. Then look straight across to the y-axis and read the number.
  • You know the value on the y-axis, such as a distance. Find it on the y-axis. Go straight across to the point. Then look straight down to the x-axis and read the number.

In this lesson, each value you look for has exactly one plotted point. Give every answer with its unit.

Try it with the graph below. Mia watched a snail crawl along a ruler. Each minute she wrote down how far it had crawled. The x-axis stands for minutes since the snail started. The y-axis stands for centimeters crawled.

Coordinate grid from x 0 to 6 and y 0 to 10, with a grid step of 1, showing 5 plotted elements. Details: point at (1, 3), named Point A; point at (2, 5), named Point B; point at (3, 6), named Point C; point at (4, 7), named Point D; point at (5, 9), named Point E.
The snail's graph: Point B is above 2 on the x-axis (2 minutes) and across from 5 on the y-axis (5 centimeters), so Point B is at (2,5)(2, 5).

Worked example

Use the graph to answer questions

Use the snail's graph above. The x-axis stands for minutes since the snail started. The y-axis stands for centimeters crawled. (a) How far had the snail crawled after 2 minutes? (b) After how many minutes had it crawled 7 centimeters? (c) How much farther had it crawled after 5 minutes than after 1 minute?

  1. Part (a) gives a time, and time is on the x-axis. Find 2 on the x-axis and go straight up to Point B. Look straight across to the y-axis and read 5. Point B is at (2, 5), so its y-coordinate, 5, is the distance in centimeters.

    (2,5) means 2 minutes, 5 centimeters(2, 5) \text{ means } 2 \text{ minutes},\ 5 \text{ centimeters}

  2. Part (b) gives a distance, and distance is on the y-axis. Find 7 on the y-axis and go straight across to Point D. Look straight down to the x-axis and read 4. Point D is at (4, 7), so its x-coordinate, 4, is the time in minutes.

    (4,7) means 4 minutes, 7 centimeters(4, 7) \text{ means } 4 \text{ minutes},\ 7 \text{ centimeters}

  3. Part (c) compares two times, so read two points. Above 5 on the x-axis is Point E, at (5, 9). Above 1 on the x-axis is Point A, at (1, 3). The y-coordinates, 9 and 3, are the two distances in centimeters.

    (5,9) and (1,3)(5, 9) \text{ and } (1, 3)

  4. Subtract the smaller distance from the larger one to find how much farther. Both numbers are centimeters, so the answer is in centimeters. On the graph, Point E is 6 units higher than Point A.

    9−3=69 - 3 = 6

Answer

After 2 minutes the snail had crawled 5 centimeters. It had crawled 7 centimeters after 4 minutes. After 5 minutes it had crawled 6 centimeters farther than after 1 minute.

Here is the big idea. On a graph of real-world data, each axis stands for one quantity with its unit, and each point pairs two values that go together. The x-coordinate is the value of the x-axis quantity. The y-coordinate is the value of the y-axis quantity. To answer a question, start from the value you know, go to its point, and read the other coordinate with its unit.

Next, you will describe the relationship a graph shows, which is how its two quantities change together. You will use it to find values that are not plotted, compare two relationships on the same plane, and find and fix graphing mistakes.

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Analyze Graphed Relationships and Fix Graphing Mistakes

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