### Graphs in the Coordinate Plane

Once we know how to plot points on a coordinate plane, we can think about how we’d go about plotting a relationship between **relation**, and there isn’t necessarily a consistent relationship between the *is* a relationship between the **function**. Remember that a function is a particular way to relate one quantity to another.

#### Graphing a Function

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

Once you know how to graph a function like this, you can simply read the relationship between the

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 **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.

#### Using Graphs to Solve Real-World Problems

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

Friends |
Presents |
Coordinates |
---|---|---|

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

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

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

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

**Graphing 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.

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

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

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

*so our y−intercept is* (0, 5).

2. Look at the coefficient multiplying the

*Every time we increase x by one, y 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

We will properly examine this last method later in this chapter!

### Examples

#### Example 1

The point (0, -2) is the boundary of which two quadrants?

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.

#### Example 2

If you move the point (-3,4) down 5, what quadrant would it be in?

Moving the point down 5 is equivalent to subtracting 5 from the y-value.

### Review

- Consider the graph of the equation
y=3 . Which quadrants does it pass through? - Consider the graph of the equation \begin{align*}y=x\end{align*}
y=x . Which quadrants does it pass through? - Consider the graph of the equation \begin{align*}y=x+3\end{align*}
y=x+3 . 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*}
ABCD . Plot them on a graph, then determine what the coordinates of the fourth point, \begin{align*}D\end{align*}D , would be. Plot that point and label it. \begin{align*}A (-4, -4) \ B (3, -4) \ C (3, 3)\end{align*}A(−4,−4) B(3,−4) C(3,3) - 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*}
x is the number of Starburst that Jaeyun gives Becky, and \begin{align*}y\end{align*}y 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*}
y in terms of \begin{align*}x\end{align*}x . - Make a table of values for \begin{align*}y\end{align*}
y with \begin{align*}x\end{align*}x -values of 0, 1, 2, 3, 4, 5. - Plot the function linking \begin{align*}x\end{align*}
x and \begin{align*}y\end{align*}y on the following scale: \begin{align*}0 \le x \le 10, \ 0 \le y \le 10\end{align*}0≤x≤10, 0≤y≤10 .

- Write an algebraic rule for \begin{align*}y\end{align*}

### Review (Answers)

To view the Review answers, open this PDF file and look for section 4.2.