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```Chapter 3
Section 3.1
Graphs
Graphs
A graph is a way to draw a "picture" of
an equation. A location on the picture is
called a point and is marked with a dot
and is a pair of numbers in parenthesis:
๐ฅ, ๐ฆ โ Point
๐ฅ ๐๐  ๐กโ๐ ๐ฅ โ ๐๐๐๐๐๐๐๐ก๐
๐ฆ ๐๐  ๐กโ๐ ๐ฆ โ ๐๐๐๐๐๐๐๐ก๐
The graph is drawn on a coordinate
system that at 2 perpendicular number
lines intersecting at the origin (0
position) of each number line. The
location is found by moving left or right
by the x-coordinate and up or down by
the y-coordinate.
7
6
B
5
D
2nd
1st
4
C
A
3
2
1
E
7
6
5
4
3
2
1
1
2
3
4
5
6
7
1
2
G
3
3rd
4th
4
F
5
6
7
D: โ3,5
C: 0,3
F: โ3, โ5
E: โ5,0
Axis
A: 5,3 B: 3,5
G: 5, โ3
Points on Equations
A point is on the graph of an equation if when you plug the x and y coordinates into the
equation you get a true equation (i.e. equal values). If the point does not work in the equation
then the point is not on the graph.
Points on the graph of 9๐ฅ 2 + ๐ฆ 2 = 25
9x2 y2 25
5
Point 1,4
9 1 2 + 4 2 = 25
9 + 16 = 25
25 = 25
4
3
2
1
5
4
3
2
1
1
1
2
3
4
5
2
3
4
5
Point 0, โ5
9 0 2 + โ5 2 = 25
0 + 25 = 25
25 = 25
Points not on the graph of 9๐ฅ 2 + ๐ฆ 2 = 25
Point 1, โ2
9 1 2 + โ2 2 = 25
9 + 4 = 25
13 = 25
Point โ2,3
9 โ2 2 + 3 2 = 25
36 + 9 = 25
45 = 25
Graphing (by Point Plotting)
One way to graph an equation is to make a table of x and y values which become
points that are plotted. Try to use enough points to find the overall shape.
y x 2
๐ฆ = ๐ฅ โ2
๐ฅ
๐ฆ
0 โ2
1 โ1
โ1 โ1
0
2
0
โ2
5
5
4
5
4
3
2
4
3
3
2
2
1
1
1
1
1
2
3
2
3
4
5
5
4
3
2
1
1
1
2
3
4
4
5
5
2
3
4
5
If the equation is solved for y continue to pick enough values for x so that as you
plot the points you can begin to understand what the shape of the graph is going
to be.
2
y x2 2x
๐ฆ = ๐ฅ โ 2๐ฅ
๐ฅ
0
1
โ1
2
3
๐ฆ
0
โ1
3
0
3
5
5
4
5
4
3
2
4
3
3
2
2
1
1
1
1
2
3
4
5
5
4
3
2
1
1
1
1
2
2
3
4
3
4
5
5
2
3
4
5
An equation that is written as ๐ฆ = ๐๐ฅ + ๐ for numbers a and b is a line.
To graph a line you only need 2 points. A third point can serve as a
check you did it right.
y 3x 1
5
5
4
๐ฆ = 3๐ฅ + 1
๐ฅ ๐ฆ
0 1
1 4
โ1 โ2
4
3
3
2
2
1
5
4
3
2
1
1
1
1
2
2
3
4
5
5
4
3
2
1
1
1
2
3
3
4
4
5
5
2
3
4
5
An equation that is written as ๐๐ฅ + ๐๐ฆ = ๐ for numbers a, b and c is also a line.
The graph can be found by filling in a table of at least 3 values. To do this you will
need to plug the values in and solve an equation.
2x 3 y 12
2๐ฅ โ 3๐ฆ = 12
๐ฅ
0
6
3
๐ฆ
โ4
0
โ2
For ๐ฅ = 0
2 0 โ 3๐ฆ = 12
0 โ 3๐ฆ = 12
โ3๐ฆ = 12
1
๐ฆ = โ 12
3
๐ฆ = โ4
8
For ๐ฆ = 0
2๐ฅ โ 3 0 = 12
2๐ฅ โ 0 = 12
2๐ฅ = 12
1
๐ฅ = 12
2
๐ฅ=6
7
6
5
4
3
2
1
8
7
6
5
4
3
2
1
1
1
2
For ๐ฅ = 3
2 3 โ 3๐ฆ = 12
6 โ 3๐ฆ = 12
โ3๐ฆ = 12 + โ6
โ3๐ฆ = 6
1
๐ฆ=โ 6
3
๐ฆ = โ2
3
4
5
6
7
8
2
3
4
5
6
7
8
```
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