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# 4.1: The Coordinate Plane

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Identify coordinates of points.
• Plot points in a coordinate plane.
• Graph a function given a table.
• Graph a function given a rule.

## Introduction

We now make our transition from a number line that stretches in only one dimension (left to right) to something that exists in two dimensions. The coordinate plane can be thought of as two number lines that meet at right angles. The horizontal line is called the \begin{align*}x-\end{align*}axis and the vertical line is the \begin{align*}y-\end{align*}axis. Together the lines are called the axes, and the point at which they cross is called the origin. The axes split the coordinate plane into four quadrants. The first quadrant (I) contains all the positive numbers from both axes. It is the top right quadrant. The other quadrants are numbered sequentially (II, III, IV) moving counter-clockwise from the first.

## Identify Coordinates of Points

When given a point on a coordinate plane, it is a relatively easy task to determine its coordinates. The coordinates of a point are two numbers - written together they are called an ordered pair. The numbers describe how far along the \begin{align*}x-\end{align*}axis and \begin{align*}y-\end{align*}axis the point is. The ordered pair is written in parenthesis, with the \begin{align*}x-\end{align*}coordinate (also called the ordinate) first and the \begin{align*}y-\end{align*}coordinate (or the ordinate) second.

\begin{align*}& (1, 7) && \text{An ordered pair with an} \ x-\text{value of one and a} \ y-\text{value of seven}\\ & (0, 5) && - \text{an ordered pair with an} \ x-\text{value of zero and a} \ y-\text{value of five}\\ & (-2.5, 4) && \text{An ordered pair with an} \ x-\text{value of}\ -2.5\ \text{and a}\ y-\text{value of four}\\ & (-107.2, -.005) && \text{An ordered pair with an} \ x-\text{value of} -107.2\ \text{and a} \ y-\text{value of}\ -0.005.\end{align*}

The first thing to do is realize that identifying coordinates is just like reading points on a number line, except that now the points do not actually lie on the number line! Look at the following example.

Example 1

Find the coordinates of the point labeled \begin{align*}P\end{align*} in the diagram to the right.

Imagine you are standing at the origin (the points where the \begin{align*}x-\end{align*}axis meets the \begin{align*}y-\end{align*}axis). In order to move to a position where \begin{align*}P\end{align*} was directly above you, you would move 3 units to the right (we say this is in the positive \begin{align*}x\end{align*} direction).

The \begin{align*}x-\end{align*}coordinate of \begin{align*}P\end{align*} is +3.

Now if you were standing at the three marker on the \begin{align*}x-\end{align*}axis, point \begin{align*}P\end{align*} would be 7 units above you (above the axis means it is in the positive \begin{align*}y\end{align*} direction).

The \begin{align*}y-\end{align*}coordinate of \begin{align*}P\end{align*} is +7.

Solution

The coordinates of point \begin{align*}P\end{align*} are (3, 7).

Example 2

Find the coordinates of the points labeled \begin{align*}Q\end{align*} and \begin{align*}R\end{align*} in the diagram to the right.

In order to get to \begin{align*}Q\end{align*} we move three units to the right, in the positive\begin{align*}-x\end{align*} direction, then two units down. This time we are moving in the negative \begin{align*}y\end{align*} direction. The \begin{align*}x\end{align*} coordinate of \begin{align*}Q\end{align*} is +3, the \begin{align*}y\end{align*} coordinate of \begin{align*}Q\end{align*} is -2.

The coordinates of \begin{align*}R\end{align*} are found in a similar way. The \begin{align*}x\end{align*} coordinate is +5 (five units in positive \begin{align*}x\end{align*}) and the \begin{align*}y-\end{align*}coordinate is again -2.

Solution

\begin{align*}Q(3, -2)\end{align*}

\begin{align*}R(5, -2)\end{align*}

Example 3

Triangle \begin{align*}ABC\end{align*} is shown in the diagram to the right. Find the coordinates of the vertices \begin{align*}A, B\end{align*} and \begin{align*}C\end{align*}.

Point \begin{align*}A\end{align*}:

\begin{align*}x-\end{align*}coordinate \begin{align*}= -2\end{align*}

\begin{align*}y-\end{align*}coordinate \begin{align*}= +5\end{align*}

Point \begin{align*}B\end{align*}:

\begin{align*}x-\end{align*}coordinate \begin{align*}= +3\end{align*}

\begin{align*}y-\end{align*}coordinate \begin{align*}= -3\end{align*}

Point \begin{align*}C\end{align*}:

\begin{align*}x-\end{align*}coordinate \begin{align*}= -4\end{align*}

\begin{align*}y-\end{align*}coordinate \begin{align*}= -1\end{align*}

Solution

\begin{align*}A (-2, 5)\! \\ B (3, -3)\! \\ C (-4, -1)\end{align*}

## Plot Points in a Coordinate Plane

Plotting points is a simple matter once you understand how to read coordinates and read the scale on a graph. As a note on scale, in the next two examples pay close attention to the labels on the axes.

Example 4

Plot the following points on the coordinate plane.

\begin{align*}A (2, 7) && B (-5, 6) && C(-6, 0) && D (-3, -3) && E (0, 2) && F (7, -5)\end{align*}

Point \begin{align*}A (2, 7)\end{align*} is 2 units right, 7 units up. It is in Quadrant I.

Point \begin{align*}B (-5, 6)\end{align*} is 5 units left, 6 units up. It is in Quadrant II.

Point \begin{align*}C (-6, 0)\end{align*} is 6 units left, 0 units up. It is on the \begin{align*}x\end{align*} axis.

Point \begin{align*}D (-3, -3)\end{align*} is 3 units left, 3 units down. It is in Quadrant III.

Point \begin{align*}E (0, 2)\end{align*} is 2 units up from the origin. It is on the \begin{align*} y\end{align*} axis.

Point \begin{align*} F (7, -5)\end{align*} is 7 units right, 5 units down. It is in Quadrant IV.

Example 5

Plot the following points on the coordinate plane.

\begin{align*} A (2.5, 0.5) && B (\pi, 1.2) && C (2, 1.75) && D (0.1, 1.2) && E (0, 0)\end{align*}

Choice of axes is always important. In Example Four, it was important to have all four quadrants visible. In this case, all the coordinates are positive. There is no need to show the negative values of \begin{align*}x\end{align*} or \begin{align*}y\end{align*}. Also, there are no \begin{align*}x\end{align*} values bigger than about 3.14, and 1.75 is the largest value of \begin{align*}y\end{align*}. We will therefore show these points on the following scale \begin{align*}\{ 0 \leq x \leq 3.5 \}\end{align*} and \begin{align*}\{ 0 \leq y \leq 2 \}\end{align*}. The points are plotted to the right.

• The tick marks on the axes do not correspond to unit increments (i.e. the numbers do not go up by one).
• The scale on the \begin{align*}x-\end{align*}axis is different than the scale on the \begin{align*}y-\end{align*}axis.
• The scale is chosen to maximize the clarity of the plotted points.

## Graph a Function Given a Table

Once we know how to plot points on a coordinate plane, we can think about how we would go about plotting a relationship between \begin{align*}x\end{align*} and \begin{align*}y\end{align*} values. So far we have been plotting sets of ordered pairs. This is called a relation, and there isn’t necessarily a relationship between the \begin{align*}x\end{align*} values and \begin{align*}y\end{align*} values. In a relation, the set of \begin{align*}x\end{align*} values is called the domain and the set of \begin{align*}y\end{align*} values is called the range. If there is a relationship between the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} values, and each \begin{align*}x\end{align*} value corresponds to exactly one \begin{align*}y\end{align*} value, then the relation is called a function. Remember that a function is a particular way to relate one quantity to another. If you read a book and can read twenty pages an hour, there is a relationship between how many hours you read and how many pages you read. You may even know that you could write the formula as either:

\begin{align*}m & = 20 \cdot h && n = \text{number of pages};\ h = \text{time measured in hours.\ OR} \ldots\\ h & = \frac{n} {20}\end{align*}

So you could use the function that related \begin{align*}n\end{align*} and \begin{align*}h\end{align*} to determine how many pages you could read in \begin{align*}3 \frac{1}{2}\end{align*} hours, or even to find out how long it took you to read forty-six pages. The graph of this function is shown right, and you can see that if we plot number of pages against number of hours, then we can simply read off the number of pages that you could read in 3.5 hours as seventy pages. You can see that in a similar way it would be possible to estimate how long it would take to read forty-six pages, though the time that was obtained might only be an approximation.

Generally, the graph of a function appears as a line or curve that goes through all points that satisfy the relationship that the function describes. If the domain of the function is all real numbers, then we call this a continuous function. However, if the domain of the function is a particular set of values (such as whole numbers), then it is called a discrete function. The graph will be a series of dots that fall along a line or curve.

In graphing equations, we assume the domain is all real numbers, unless otherwise stated. Often times though, when we look at data in a table, the domain will be whole numbers (number of presents, number of days, etc.) and the function will be discrete. Sometimes the graph is still shown as a continuous line to make it easier to interpret. Be aware of the difference between discrete and continuous functions as you work through the examples.

Example 6

Sarah is thinking of the number of presents she receives as a function of the number of friends who come to her birthday party. She knows she will get a present from her parents, one from her grandparents and one each from her uncle and aunt. She wants to invite up to ten of her friends, who will each bring one present. She makes a table of how many presents she will get if one, two, three, four or five friends come to the party. Plot the points on a coordinate plane and graph the function that links the number of presents with the number of friends. Use your graph to determine how many presents she would get if eight friends show up.

Number of Friends Number of Presents
0 4
1 5
2 6
3 7
4 8
5 9

The first thing we need to do is decide how our graph should appear. We need to decide what the independent variable is, and what the dependant variable is. Clearly in this case, the number of friends can vary independently (the domain). The number of presents must depend on the number of friends who show up (the range).

We will therefore plot friends on the \begin{align*}x-\end{align*}axis and presents on the \begin{align*}y-\end{align*}axis. Let's add another column to our table containing the coordinates that each (friends, presents) ordered pair gives us.

No. of friends \begin{align*}(x)\end{align*} no. of presents \begin{align*}(y)\end{align*} coordinates \begin{align*}(x, y)\end{align*}
0 4 (0, 4)
1 5 (1, 5)
2 6 (2, 6)
3 7 (3, 7)
4 8 (4, 8)
5 9 (5, 9)

Next we need to set up our axes. It is clear that the number of friends and number of presents both must be positive, so we do not need to worry about anything other than Quadrant I. We need to choose a suitable scale for the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} axes. We need to consider no more than eight friends (look again at the question to confirm this), but it always pays to allow a little extra room on your graph. We also need the \begin{align*}y\end{align*} scale to accommodate the presents for eight people. We can see that this is still going to be under 20!

The scale of the graph on the right shows room for up to 12 friends and 15 presents. This will be fine, but there are many other scales that would be equally as good!

Now we proceed to plot the points. The first five points are the coordinates from our table. You can see they all lay on a straight line, so the function that describes the relationship between \begin{align*}x\end{align*} and \begin{align*}y\end{align*} will be linear. To graph the function, we simply draw a line that goes through all five points. This line represents the function.

This is a discrete problem since Sarah can only invite a whole numbers of friends. For instance, it would be impossible for 2.4 friends to show up. Keep in mind that the only permissible points for the function are those points on the line which have integer \begin{align*}x\end{align*} and \begin{align*}y\end{align*} values.

The graph easily lets us find other values for the function. For example, the question asks how many presents Sarah would get if eight friends come to her party. Don't forget that \begin{align*}x\end{align*} represents the number of friends and \begin{align*}y\end{align*} represents the number of presents. If we look at \begin{align*}x=8\end{align*} we can see that the function has a \begin{align*}y\end{align*} value of 12.

Solution

If 8 friends show up, Sarah will receive a total of 12 presents.

## Graph a Function Given a Rule

If we are given a rule instead of a table, we can proceed to graph the function in one of two ways. We will use the following example to show each way.

Example 7

Ali is trying to work out a trick that his friend showed him. His friend started by asking him to think of a number. Then double it. Then add five to what he got. Ali has written down a rule to describe the first part of the trick. He is using the letter \begin{align*}x\end{align*} to stand for the number he thought of and the letter \begin{align*}y\end{align*} to represent the result of applying the rule. He wrote his rule in the form of an equation.

\begin{align*}y=2x+5\end{align*}

Help him visualize what is going on by graphing the function that this rule describes.

## Method One - Construct a Table of Values

If we wish to plot a few points to see what is going on with this function, then the best way is to construct a table and populate it with a few \begin{align*}x, y\end{align*} pairs. We will use 0, 1, 2 and 3 for \begin{align*}x\end{align*} values.

\begin{align*}x\end{align*} \begin{align*}y\end{align*}
0 \begin{align*}2\cdot 0 + 5 = 0 + 5 = 5\end{align*}
1 \begin{align*}2\cdot 1+ 5 = 2 + 5 = 7\end{align*}
2 \begin{align*}2\cdot 2 + 5 = 4 + 5 = 9\end{align*}
3 \begin{align*}2\cdot 3 + 5 = 6 + 5 = 11\end{align*}

Next, we plot the points and join them with our line.

This method is nice and simple. Plus, with linear relationships there is no need to plot more than two or three points. In this case, the function is continuous because the domain (the number Ali is asked to think of) is all real numbers, even though he may only be thinking of positive whole numbers.

## Method Two - Intercept and Slope

One other way to graph this function (and one that we will learn in more detail in the next lesson) is the slope-intercept method. To do this, follow the following steps:

1. Find the \begin{align*}y\end{align*} value when \begin{align*}x=0\end{align*}.

\begin{align*}y(0) = 2\cdot 0 + 5 = 5\end{align*} So our \begin{align*}y-\end{align*}intercept is (0, 5)

2. Look at the coefficient multiplying the \begin{align*}x\end{align*}.

Every time we increase \begin{align*}x\end{align*} by one, \begin{align*}y\end{align*} increases by two so our slope is +2.

3. Plot the line with the given slope that goes through the intercept. We start at the point (0, 5) and move over one in the \begin{align*}x\end{align*} direction, then up two in the \begin{align*}y\end{align*} direction. This gives the slope for our line, which we extend in both directions.

We will properly examine this last method in the next lesson!

## Lesson Summary

• The coordinate plane is a two-dimensional space defined by a horizontal number line (the \begin{align*}x-\end{align*}axis) and a vertical number line (the \begin{align*}y-\end{align*}axis). The origin is the point where these two lines meet. Four areas, or quadrants, are formed as shown in the diagram at right.
• Each point on the coordinate plane has a set of coordinates, two numbers written as an ordered pair which describe how far along the \begin{align*}x-\end{align*}axis and \begin{align*}y-\end{align*}axis the point is. The \begin{align*}x-\end{align*}coordinate is always written first, then the \begin{align*}y-\end{align*}coordinate. Here is an exaxmple \begin{align*}(x, y)\end{align*}.
• Functions are a way that we can relate one quantity to another. Functions can be plotted on the coordinate plane.

## Review Questions

1. Identify the coordinates of each point, \begin{align*}A - F\end{align*}, on the graph to the right.
2. Plot the following points on a graph and identify which quadrant each point lies in:
1. (4, 2)
2. (-3, 5.5)
3. (4, -4)
4. (-2, -3)
3. The following three points are three vertices of square \begin{align*}ABCD\end{align*}. Plot them on a graph then determine what the coordinates of the fourth point, \begin{align*}D\end{align*}, would be. Plot that point and label it. \begin{align*}A (-4, -4)\! \\ B (3, -4)\! \\ C (3, 3)\end{align*}
4. Becky has a large bag of M&Ms that she knows she should share with Jaeyun. Jaeyun has a packet of Starburst. Becky tells Jaeyun that for every Starburst he gives her, she will give him three M&Ms in return. If \begin{align*}x\end{align*} is the number of Starburst that Jaeyun gives Becky, and \begin{align*}y\end{align*} is the number of M&Ms he gets in return then complete each of the following.
1. Write an algebraic rule for \begin{align*}y\end{align*} in terms of \begin{align*}x\end{align*}
2. Make a table of values for \begin{align*}y\end{align*} with \begin{align*}x\end{align*} values of 0, 1, 2, 3, 4, 5.
3. Plot the function linking \begin{align*}x\end{align*} and \begin{align*}y\end{align*} on the following scale \begin{align*}\{ 0 \leq x \leq 10 \}, \{0 \leq y \leq 10 \}\end{align*}.

1. \begin{align*}A (5, 6) \quad B (-5, 5) \quad C (-2, 3) \quad D (-2, -2) \quad E(3, -4) \quad F(2, -6)\end{align*}
2. (a) \begin{align*}y = 3x\end{align*}

(b)

\begin{align*}x\end{align*} \begin{align*}y\end{align*}
0 0
1 3
2 6
3 9
4 12
5 15

(c)

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