4.1: The Coordinate Plane
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
Lydia lives 2 blocks north and one block east of school; Travis lives three blocks south and two blocks west of school. What’s the shortest line connecting their houses?
The Coordinate Plane
We’ve seen how to represent numbers using number lines; now we’ll see how to represent sets of numbers using a coordinate plane. 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, which are numbered sequentially (I, II, III, IV) moving counter-clockwise from the upper right.
Identify Coordinates of Points
When given a point on a coordinate plane, it’s easy 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 parentheses, with the \begin{align*}x-\end{align*}coordinate (also called the abscissa) 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} \ -.005\end{align*}
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 above
Solution
Imagine you are standing at the origin (the point 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 3 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.
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.
Solution
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 the positive \begin{align*}x-\end{align*}direction) and the \begin{align*}y-\end{align*}coordinate is again −2.
The coordinates of \begin{align*}Q\end{align*} are (3, −2). The coordinates of \begin{align*}R\end{align*} are (5, −2).
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-\text{coordinate} = -2\end{align*}
\begin{align*}y-\text{coordinate} = +5\end{align*}
Point \begin{align*}B\end{align*}:
\begin{align*}x-\text{coordinate} = +3\end{align*}
\begin{align*}y-\text{coordinate} = -3\end{align*}
Point \begin{align*}C\end{align*}:
\begin{align*}x-\text{coordinate} = -4\end{align*}
\begin{align*}y-\text{coordinate} = -1\end{align*}
Solution
\begin{align*}A(-2, 5)\end{align*}
\begin{align*}B(3, -3)\end{align*}
\begin{align*}C(-4, -1)\end{align*}
Plot Points in a Coordinate Plane
Plotting points is simple, 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) \quad B(-4, 6) \quad D(-3, -3) \quad E(0, 2)\quad 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(-4, 6)\end{align*} is 4 units left, 6 units up. It is in Quadrant II.
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 right on the \begin{align*}y-\end{align*}axis, between Quadrants I and II.
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) \quad B(\pi, 1.2) \quad C(2, 1.75) \quad D(0.1, 1.2) \quad E(0, 0)\end{align*}
Here we see the importance of choosing the right scale and range for the graph. In Example 4, our points were scattered throughout the four quadrants. In this case, all the coordinates are positive, so we don’t 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 can therefore show just the part of the coordinate plane where \begin{align*}0 \le x \le 3.5\end{align*} and \begin{align*}0 \le y \le 2\end{align*}.
Here are some other important things to notice about this graph:
- The tick marks on the axes don’t correspond to unit increments (i.e. the numbers do not go up by one each time). This is so that we can plot the points more precisely.
- The scale on the \begin{align*}x-\end{align*}axis is different than the scale on the \begin{align*}y-\end{align*}axis, so distances that look the same on both axes are actually greater in the \begin{align*}x-\end{align*}direction. Stretching or shrinking the scale in one direction can be useful when the points we want to plot are farther apart in one direction than the other.
For more practice locating and naming points on the coordinate plane, try playing the Coordinate Plane Game at http://www.math-play.com/Coordinate%20Plane%20Game/Coordinate%20Plane%20Game.html.
Graph a Function Given a Table
Once we know how to plot points on a coordinate plane, we can think about how we’d go about plotting a relationship between \begin{align*}x-\end{align*}and \begin{align*}y-\end{align*}values. So far we’ve just been plotting sets of ordered pairs. A set like that is a relation, and there isn’t necessarily a relationship between the \begin{align*}x-\end{align*}values and \begin{align*}y-\end{align*}values. 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’re reading 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*}n=20h\end{align*} or \begin{align*}h= \frac{n} {20} \end{align*}, where \begin{align*}h\end{align*} is the number of hours you spend reading and \begin{align*}n\end{align*} is the number of pages you read. To find out, for example, how many pages you could read in \begin{align*}3 \frac{1}{2}\end{align*} hours, or how many hours it would take you to read 46 pages, you could use one of those formulas. Or, you could make a graph of the function:
Once you know how to graph a function like this, you can simply read the relationship between the \begin{align*}x-\end{align*}and \begin{align*}y-\end{align*}values off the graph. You can see in this case that you could read 70 pages in \begin{align*}3 \frac{1}{2}\end{align*} hours, and it would take you about \begin{align*}2 \frac{1}{3}\end{align*} hours to read 46 pages.
Generally, the graph of a function appears as a line or curve that goes through all points that have the relationship that the function describes. If the domain of the function (the set of \begin{align*}x-\end{align*}values we can plug into the function) is all real numbers, then we call it a continuous function. If the domain of the function is a particular set of values (such as whole numbers only), then it is called a discrete function. The graph will be a series of dots, but they will still often fall along a line or curve.
In graphing equations, we assume the domain is all real numbers, unless otherwise stated. Often, 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. But sometimes we’ll still draw the graph 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, but the number of presents must depend on the number of friends who show up.
So we’ll 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.
Friends \begin{align*}(x)\end{align*} | 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 only need to show points in Quadrant I. Now we need to choose a suitable scale for the \begin{align*}x-\end{align*} and \begin{align*}y-\end{align*}axes. We only need to consider 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 this graph has room for up to 12 friends and 15 presents. This will be fine, but there are many other scales that would be equally good!
Now we proceed to plot the points. The first five points are the coordinates from our table. You can see they all lie 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 positive whole number of friends. For instance, it would be impossible for 2.4 or -3 friends to show up. So although the line helps us see where the other values of the function are, the only points on the line that actually are values of the function are the ones with positive whole-number coordinates.
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 the graph where \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 either 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 the result. 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 final 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’ll use 0, 1, 2 and 3 for \begin{align*}x-\end{align*}values.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
0 | 5 |
1 | 7 |
2 | 9 |
3 | 11 |
Next, we plot the points and join them with a line.
This method is nice and simple—especially with linear relationships, where we don’t need to plot more than two or three points to see the shape of the graph. In this case, the function is continuous because the domain is all real numbers—that is, Ali could think of any real number, even though he may only be thinking of positive whole numbers.
Method Two - Intercept and Slope
Another way to graph this function (one that we’ll learn in more detail in a later lesson) is the slope-intercept method. To use this method, follow these steps:
1. Find the \begin{align*}y\end{align*} value when \begin{align*}y = 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 later in this chapter!
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 above.
- 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, in the form \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
- Identify the coordinates of each point, \begin{align*}A-F\end{align*}, on the graph below.
- Draw a line on the above graph connecting point \begin{align*}B\end{align*} with the origin. Where does that line intersect the line connecting points \begin{align*}C\end{align*} and \begin{align*}D\end{align*}?
- Plot the following points on a graph and identify which quadrant each point lies in:
- (4, 2)
- (-3, 5.5)
- (4, -4)
- (-2, -3)
- Without graphing the following points, identify which quadrant each lies in:
- (5, 3)
- (-3, -5)
- (-4, 2)
- (2, -4)
- Consider the graph of the equation \begin{align*}y=3\end{align*}. Which quadrants does it pass through?
- Consider the graph of the equation \begin{align*}y=x\end{align*}. Which quadrants does it pass through?
- Consider the graph of the equation \begin{align*}y=x+3\end{align*}. Which quadrants does it pass through?
- The point (4, 0) is on the boundary between which two quadrants?
- The point (0, -5) is on the boundary between which two quadrants?
- If you moved the point (3, 2) five units to the left, what quadrant would it be in?
- 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*}
- In what quadrant is the center of the square from problem 10? (You can find the center by drawing the square’s diagonals.)
- What point is halfway between (1, 3) and (1, 5)?
- What point is halfway between (2, 8) and (6, 8)?
- What point is halfway between the origin and (10, 4)?
- What point is halfway between (3, -2) and (-3, 2)?
- 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.
- Write an algebraic rule for \begin{align*}y\end{align*} in terms of \begin{align*}x\end{align*}.
- 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.
- Plot the function linking \begin{align*}x\end{align*} and \begin{align*}y\end{align*} on the following scale: \begin{align*}0 \le x \le 10, \ 0 \le y \le 10\end{align*}.
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