4.2: Graphs in the Coordinate Plane
What if you were given a function rule like \begin{align*}y = 2x - 3\end{align*}
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CK-12 Foundation: 0402S Graphs in the Coordinate Plane (H264)-1
Guidance
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.
Example A
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 B
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 C
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!
Watch this video for help with the Examples above.
CK-12 Foundation: Graphs in the Coordinate Plane
Vocabulary
- 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.
Guided Practice
1. The point (0, -2) is the boundary of which two quadrants?
2. If you move the point (-3, 4) down 5, what quadrant would it be in?
Solutions:
1. Since the x-value is 0, the point is on the y-axis. Since the y-value is negative, the point is on the lower half of the y-axis. This is the boundary between the 3rd and 4th quadrants.
2. Moving the point down 5 is equivalent to subtracting 5 from the y-value. \begin{align*}(-3, 4-5)=(-3, -1)\end{align*}. Since both coordinates are now negative, this is in the 3rd quadrant.
Practice
- 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*}.
Image Attributions
Here you'll learn how to make a table of values and graph a function given the function's rule.