Functions and Graphs

STEM-Plus Content offers K-12 math and other STEM content in both English and Japanese.
The content aids in vocabulary acquisition for both English and Japanese learners.
How does it achieve this?
Through the content, learners grasp words and phrases more intuitively because they already know their meanings in their native language.
It is like learning the word “apple” by looking at a picture of one.
STEM-Plus Content facilitates easy learning of abstract yet useful words.

To ensure effective and efficient learning, the content is designed to be concise and illustrative.
Learners can easily understand the material through images and examples.

In addition, the content provides useful tips on math and other STEM subjects, supporting
academic success in these important fields.

How to use:
Carefully selected English words and phrases related to “Functions and Graphs on the Coordinated Planes” are highlighted in yellow in the images below. The words are also summarized at the bottom of this page.
We recommend reviewing them regularly during your daily math studies to enhance your vocabulary.

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  • An ordered pair is two numbers or letters (or anything else) written in the form (x, y).
    順序対とは (x, y) の形式で書かれたふたつの数字または文字(またはその他何でも)のこと。
  • A relation is a set of ordered pairs.
  • The domain is the set of all the first elements (or x-values) of each ordered pair.
  • The range is the set of all the second elements (or y-values) of each ordered pair.
Definition and example of relation, domain, and range



  • A relation is a function if there is exactly one y-value for each x-value.
  • You can think of a function as an input-output machine.
Definition and examples of function


Coordinate Planes

  • A coordinate plane is formed by a horizontal and vertical number line, called coordinate axes.
  • The horizontal axis is the x-axis.
  • The vertical axis is the y-axis.
  • The axes meet at a point called the origin.
Definition of coordinate plane

Coordinate Plane

Points in a Coordinate Plane

  • Each point in a coordinate plane is represented by an ordered pair (x, y).
  • The first number is the x-coordinate, and the second number is the y-coordinate.
  • The coordinates are written as (x, y).
Illustrating how to graph points in a coordinate plane

Graphing Points in a Coordinate Plane


relate 関係がある
 /relate A to B: AをBと関係づける
📎relation {複数の物の間の}関係

domain {扱う}領域、分野

range 範囲

exact 正確な 📎exactly 正確に

function 機能;関数

coordinate {うまく行くよう活動など}を調整する;{服など}を合わせる
 /重要度が同じな ⇔📎subordinate 下位の

origin 起源

vertical 垂直な ⇔📎horizontal 水平な

represent を代表する;に相当する;を表す



“Coordinate (Planes)”とcoordinate

Please remember that “coordinate” can mean both “adjust” and “Coordinate (Plane).”

By the way, the reason we use “coordinate” to refer to “Coordinate (Planes)” is not because x and y are arranged or adjusted well. It’s because x and y have the same level of importance, which is the same as the adjective meaning of “coordinate.” Here are the details:

Origin and meaning of “Coordinate (Planes)”

“Leibniz in letters of 1694 first used the word “coordinates” in the strictly modern sense and recognized the two coordinates as on the same footing.” (*1)

The two coordinates refer to a certain type of auxiliary line “abscissa” and “ordinate”(*2) used in geometry problems.
The modern sense means that “both coordinates are equally strong and together describe a point in the plane or space.”(*3)
So, the “co” in “Coordinate (Planes)” means “joint” as in coauthor and cofounder.

*1:Boyer, Barl B (2004 ). History of Analytic Geometry. Dover Publications, Inc: Mineola, New York. p133
*2: Abscissa means “horizontal line” and refers to a cut line, which became the basis of x-coordinates.
Ordinate means “vertical line” and refers to a parallel line, which became the basis of y-coordinates.
*3:Sofia Neovius (2013). René Descartes’ Foundations of Analytic Geometry and Classification of Curves p20


Other tips

We couldn’t find an appropriate Japanese translation for “mapping diagram.”
Since “mapping” can refer to cases other than “写像” (like we have shown in the above example), we have arbitrarily used “対応図” to describe it.

In Japan, “the range” is not introduced until high school. Even in the third year of junior high school, we use the term “the domain of y-values” instead.

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