5.1 Graphing Basics

Leanne Thompson

5.1 Graphing Basics

The Coordinate Plane

Just like maps use a grid system to identify locations, a grid system is used in algebra to show a relationship between two variables within a rectangular coordinate system. The rectangular coordinate system is called the coordinate plane.

Described in previous paragraphs. Top right quadrant labelled “I”, top left “II”, bottom left “III”, and bottom right “IV”. The horizontal number line within the coordinate plane is called the x-axis, and the vertical number line within the coordinate plane is called the y-axis. These axes divide the coordinate plane into four regions, called quadrants. The quadrants are identified by Roman numerals, beginning on the upper right and proceeding counterclockwise. If a point is on one of the axes, we say that it is not considered to be in a quadrant. The quadrants can be seen in the provided figure.

Within the coordinate plane, every point is represented by an ordered pair in the form (x, y). The first number in the ordered pair is the x-coordinate of the point, and the second number is the y-coordinate of the point. The ordered pair of the point where the axes cross has both coordinates equal to zero, so its ordered pair is (0, 0). The point (0, 0) has a special name. It is called the origin.

Plotting Points on the Coordinate Plane

Figure 2. The result of the process described in previous paragraph plotting the point (1,3). We use the coordinates of an ordered pair to determine where the point should be plotted on the coordinate plane. Let’s plot the point (1, 3) as an example. To plot this point, first locate 1 on the x-axis and lightly sketch a vertical line through x = 1. Next, locate 3 on the y-axis and sketch a vertical line through y = 3. Now, find the point where these two lines meet—that is the point with coordinates (1, 3). The provided figure shows the plotted point.

Practice 1

Plot each point on the coordinate planes below. If possible, identify the quadrant in which the point is located. Label each point with the assigned letter. Make sure that you note the scale of each axis before you start plotting your points.

a)

Blank coordinate plane. Both axes have a minimum of -6 and maximum of 6 and a scale of 1.

A (–5, 4)
B (–3, –4)
C (2, –3)
D (–2, 3)
E (3, $\frac{5}{2}$ )

b)

Blank coordinate plane. Both axes have a minimum of -6 and maximum of 6 and a scale of 2.

A (0, 5)
B (4, 0)
C (–3, 0)
D (0, 0)
E (0, –1)

Practice 2

Identify the ordered pair of each point shown on the coordinate planes.

a)

Each point is described in reference to the origin. A is 3 left and 3 up. B is 1 left and 3 down. C is 2 right and 4 up. D is 4 right and 4 down. E is 2 down. F is 3 right.

b)

A graph plotting the points A (0, negative 2), B (negative 2, 0), C (0, 5), D (5, 0).

Creating a Table of Values

To graph an equation on the coordinate plane, we first need to identify the points we will be plotting. To identify some points that correspond to the equation, we can create a table of values. To create a table of values, choose some values for x, then solve the equation for y. You will want to choose values for x that will work with the coordinate plane you are using or make sense for the context of the question. Alternatively, you could also pick values for y, and then solve for x; however, this is less common.

Example 1

Complete the table of values for the equation $y=4x-2$ .

x	y	(x, y)
–1
0
1
2

Table 5.1.1

Steps

Solution

Calculate the y values using the provided x values.

x = 2 y = 4(2) – 2 y = 6	x = 0 y = 4(0) – 2 y = –2
x = 1 y = 4(1) – 2 y = 2	x = –1 y = 4(–1) – 2 y = –6

Fill in the table of values.

x	y	(x, y)
–1	–6	(–1, –6)
0	–2	(0, –2)
1	2	(1, 2)
2	6	(2, 6)

Practice 3

Fill in the table of values for the equations.

a) $y=3x-1$

x

y

(x, y)

–3

–1

1

3

b) $y=-3x+2$

x

y

(x, y)

–5

14

11

–2

c) $5x-4y=20$

x

y

(x, y)

–4

–5

4

8

Graphing an Equation

We can now graph an equation by creating a table of values, and then plotting these points on a coordinate plane. It is true that it only takes two points to determine a line, but it is a good habit to use three points. If you plot only two points and one of them is incorrect, you can still draw a line, but it will not represent the solutions to the equation—it will be the wrong line. If you use three points, and one is incorrect, the points will not line up. This will tell you that something is wrong, and you need to check your work.

Example 2

Graph $y=2x-3$ .

Table 5.1.2

Steps

Solution

Create a table of values. Use the x- or y-values provided, or if they have not been provided, choose at least three x-values.

x	y	(x,y)
–1	–5	(–1, –5)
1	–1	(1, –1)
3	3	(3, 3)

Plot your points on the coordinate plane. Connect your points with a line.

Graphs the line 2x−3 with points at (-1,-5), (1,-1), and (3,3).

Practice 4

Graph the following equations.

a) $y=3x-1$

b) $y=\frac{1}{2}x+3$

x

y

(x, y)

–

x	y	(x, y)
-6
-2
2

Homework

Plot each point on the coordinate planes below. If possible, identify the quadrant in which the point is located. Label each point with the assigned letter or with its ordered pair.

a)
a (–2, 1)
b (–3, –1)
c (4, –4)
d (–4, 4)
e (–4, $\frac{3}{2}$ ) b)
a (–5, 0)
b (3, 0)
c (0, 0)
d (0, –1)
e (0, 4)
Identify the ordered pair of each point shown on the coordinate planes below.

a)
b)
Complete the table of values for the equations.

a) $y=2x+4$

x y (x, y)

0

1

2

3

b) $y=3x-1$

x y (x, y)

–2

–1

0

2

c) $y=-x+5$

x y (x, y)

0

2

1

6

d) $y=-x-2$

x y (x, y)

–5

–3

–1

–3

e) $y=\frac{1}{3}x+1$

x y (x, y)

–3

0

3

6

f) $y=-\frac{3}{2}x-2$

x y (x, y)

2

4

6

–14

g) $x+3y=6$

x y (x, y)

6

9

–2

15

h) $3x-4y=12$

x y (x, y)

–6

–2

0

2
Graph the following equations.

a) $y=2x-3$

b) $y=-4x$

c) $y=x$

d) $y=\dfrac{1}{3}x-1$

e) $2x+y=2$

f) $4x+2y=8$

g) $y=3x-1$

h) $y=-3x+3$

i) $y=-x-3$

j) $y=\dfrac{1}{2}x+2$

k) $x+y=-3$

l) $-3x+y=7$
Which ordered pairs are solutions to $y=6x-10$ ? Circle all that apply.

a) (–1, 16)
b) (0, –5)
c) (4, 14)
d) (6, 20)
Given the equation $y=6x-10$ , what is the value of x in the ordered pair (x, –16)?

a) 10
b) –1
c) 5
d) –8
Is the point (–3, 0) on the x-axis or the y-axis? How do you know?