Chapter 12 · The coordinate plane

Analyze Graphed Relationships and Fix Graphing Mistakes

Read the pattern in a graph and use it to find values that are not plotted. Compare two relationships on one plane, then catch and fix graphing mistakes.

15 min read

You already know how to plot points from a rule and how to say what one point means. Now you will read a whole graph at once.

A graph can show a relationship. A relationship is the way one quantity changes when another quantity changes. A quantity is something you can count or measure, like weeks or dollars. For example, the dollars you have saved change as the weeks go by.

In this lesson you will:

describe the relationship a graph shows and use it to find a value that is not plotted,
compare two relationships on the same graph, and
find and fix graphing mistakes.

Key concept

The pattern is the relationship

Each point pairs an $x$ -value (how far across) with a $y$ -value (how far up). To see the relationship, look at how the points change from one to the next and ask one question: each time $x$ goes up by 1, how much does $y$ change?

In every relationship in this lesson, $y$ changes by the same amount each time $x$ goes up by 1. That steady change is the pattern. Once you know a steady pattern, you can use it to find values that are not plotted (as long as the pattern continues), to compare two relationships, and to spot a point that does not fit.

Maya and Leo each save money every week. The graph below shows both of them on the same coordinate plane.

The $x$ -axis (across) shows the number of weeks.
The $y$ -axis (up) shows the dollars saved.
Maya is Points A, B, C and D, at $(1, 3)$ , $(2, 6)$ , $(3, 9)$ and $(4, 12)$ .
Leo is Points E, F, G and H, at $(1, 1)$ , $(2, 2)$ , $(3, 3)$ and $(4, 4)$ .

Read one point to warm up. Point B is at $(2, 6)$ . Its two numbers are its coordinates. They say that after 2 weeks, Maya has saved 6 dollars.

Points A to D show Maya and Points E to H show Leo. Each grid step across is 1 week, and each grid step up is 1 dollar.

Worked example

Describe the pattern and go past the graph

Use Maya's points on the graph: Point A $(1, 3)$ , Point B $(2, 6)$ , Point C $(3, 9)$ and Point D $(4, 12)$ . How does $y$ change each time $x$ goes up by 1? If Maya keeps saving the same way, how many dollars will she have after 6 weeks?

Go from each of Maya's points to the next one. Each move is 1 week to the right, so x goes up by 1. Subtract the y-values to see how far up each move goes.
$6 - 3 = 3 \qquad 9 - 6 = 3 \qquad 12 - 9 = 3$
The change is the same every time, so you can describe the relationship: each time x goes up by 1, y goes up by 3. In the story, that means Maya saves 3 dollars each week.
Week 6 is not plotted, so use the pattern. Start at Point D, where week 4 has 12 dollars. Week 6 is 2 more weeks, so add 3 dollars two times. This is valid because the problem says Maya keeps saving the same way.
$12 + 3 = 15 \qquad 15 + 3 = 18$
Check with a rule, which tells you how to get y from x. In each of Maya's points, the y-value is 3 times the x-value: 3 is 3 times 1, 6 is 3 times 2, 9 is 3 times 3 and 12 is 3 times 4. Use the same rule for week 6 and you get the same answer.
$y = 3 \times x \qquad 3 \times 6 = 18$

Answer

Each time $x$ goes up by 1, $y$ goes up by 3, so Maya saves 3 dollars each week. If she keeps saving the same way, she will have 18 dollars after 6 weeks. That is the point $(6, 18)$ .

Worked example

Compare two relationships on one graph

On the same graph, Maya is Points A to D and Leo is Points E to H. Who saves faster? After 4 weeks, how much more has Maya saved than Leo? Does the gap between them stay the same?

Find Leo's pattern the same way you found Maya's. His points are (1, 1), (2, 2), (3, 3) and (4, 4). Subtract the y-values from one point to the next. Each time x goes up by 1, Leo's y-value goes up by 1, so he saves 1 dollar each week.
$2 - 1 = 1 \qquad 3 - 2 = 1 \qquad 4 - 3 = 1$
Compare the two changes. Each week, Maya's y-value goes up by 3 and Leo's goes up by 1. Both sets of points use the same axes, so you can also see it: Maya's points rise more for each step to the right.
$3 > 1$
Compare the same week. Week 4 is x = 4, so read the two points above 4 on the x-axis. Point D is (4, 12) and Point H is (4, 4). Subtract Leo's dollars from Maya's dollars.
$12 - 4 = 8$
Check the gap in the other weeks. It is 2 dollars at week 1, 4 at week 2, 6 at week 3 and 8 at week 4. On the graph, the two sets of points get farther apart as you move right. The gap grows by 2 each week because Maya adds 3 dollars while Leo adds 1.
$3 - 1 = 2 \qquad 6 - 2 = 4 \qquad 9 - 3 = 6 \qquad 12 - 4 = 8$

Answer

Maya saves faster: her $y$ -value goes up by 3 each week and Leo's goes up by 1. After 4 weeks Maya has 12 dollars and Leo has 4 dollars, so Maya has saved 8 dollars more. The gap does not stay the same. It grows by 2 dollars each week.

A graph only helps if every point is in the right place. You can check a graph the same way you read one: compare each point with the rule or the table it came from. A correctly plotted point matches one pair in the table exactly. A point that does not match needs a closer look.

Here are three mistakes to watch for.

Three graphing mistakes and how to fix them

Mistake	Clue	Fix
Swapped coordinates	The point's pair is not in the table, but the same two numbers in the other order are.	Swap the numbers back. Go across by the $x$ -value first, then up by the $y$ -value.
A point that breaks the rule	The $x$ -value is in the table, but the $y$ -value is not the one the rule gives.	Use the rule to find the right $y$ -value. Move the point up or down to it.
An axis read as the other quantity	A reading gives each number the wrong quantity, such as weeks for dollars.	Match the first number to the $x$ -axis quantity and the second number to the $y$ -axis quantity.

Priya saves 2 dollars each week. Her rule is $y = 2 \times x$ , where $x$ is the number of weeks and $y$ is the dollars saved. She made a correct table for weeks 1 to 5. Then she drew the graph below, with weeks on the $x$ -axis and dollars on the $y$ -axis. She plotted one point for each week, in order from Point A for week 1 to Point E for week 5.

Some of her points are in the wrong place. Use the rule to think about where each point should be, and try to spot the mistakes before you read on.

Priya's graph, with weeks across and dollars up. She plotted Point A at

(1, 2)

, Point B at

(2, 4)

, Point C at

(6, 3)

, Point D at

(4, 7)

and Point E at

(5, 10)

Worked example

Find and fix the mistakes in Priya's graph

Priya's rule is $y = 2 \times x$ . Her table for weeks 1 to 5 gives the pairs $(1, 2)$ , $(2, 4)$ , $(3, 6)$ , $(4, 8)$ and $(5, 10)$ . Her graph has Points A to E at $(1, 2)$ , $(2, 4)$ , $(6, 3)$ , $(4, 7)$ and $(5, 10)$ . Which points are in the wrong place, and where do they belong?

Say what every correct point must do. Each point must sit on its own pair from the table: Point A on the first pair, Point B on the second, and so on. Every table pair fits the rule, so the y-value is 2 times the x-value.
$y = 2 \times x$
Check Points A, B and E. Each one sits exactly on its table pair, and each pair makes the rule true. These three points are correct.
$2 \times 1 = 2 \qquad 2 \times 2 = 4 \qquad 2 \times 5 = 10$
Check Point C. It sits at (6, 3), but its table pair is the third one, (3, 6). The rule agrees that (6, 3) is wrong: 2 times 6 is 12, not 3. The two numbers are swapped. The first number tells how far to go across, but Priya went 6 across and 3 up. Move C to 3 across and 6 up.
$2 \times 6 = 12 \neq 3 \qquad (6, 3) \rightarrow (3, 6)$
Check Point D. It sits at (4, 7), but its table pair is the fourth one, (4, 8). The x-value 4 is right. The y-value breaks the rule: 2 times 4 is 8, not 7. Swapping would not fix it, because (7, 4) is not a table pair. Move D up 1 to (4, 8).
$2 \times 4 = 8 \neq 7 \qquad (4, 7) \rightarrow (4, 8)$
Check the fixed graph. The points are now (1, 2), (2, 4), (3, 6), (4, 8) and (5, 10). From each point to the next, x goes up by 1 and y goes up by 2, so all five points fit the pattern.
$4 - 2 = 2 \qquad 6 - 4 = 2 \qquad 8 - 6 = 2 \qquad 10 - 8 = 2$

Answer

Points A, B and E are correct. Point C has swapped coordinates and belongs at $(3, 6)$ . Point D breaks the rule and belongs at $(4, 8)$ .

The big idea: in the relationships you saw here, $y$ changes by the same amount each time $x$ goes up by 1. Find that change, and you can do three things:

find a value that is not plotted, as long as the pattern continues,
compare two relationships on the same plane, and
check every point against its rule or table, then fix any point that does not match.

When you read a point, keep the axes straight. The first number goes with the $x$ -axis quantity, and the second number goes with the $y$ -axis quantity.

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