Download Do 3.1

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:
𝑥, 𝑦 − Point
𝑥 𝑖𝑠 𝑡ℎ𝑒 𝑥 − 𝑐𝑜𝑜𝑑𝑖𝑛𝑎𝑡𝑒
𝑦 𝑖𝑠 𝑡ℎ𝑒 𝑦 − 𝑐𝑜𝑜𝑑𝑖𝑛𝑎𝑡𝑒
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: −3,5
C: 0,3
F: −3, −5
E: −5,0
Axis
A: 5,3 B: 3,5
G: 5, −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 9𝑥 2 + 𝑦 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, −5
9 0 2 + −5 2 = 25
0 + 25 = 25
25 = 25
Points not on the graph of 9𝑥 2 + 𝑦 2 = 25
Point 1, −2
9 1 2 + −2 2 = 25
9 + 4 = 25
13 = 25
Point −2,3
9 −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
𝑦 = 𝑥 −2
𝑥
𝑦
0 −2
1 −1
−1 −1
0
2
0
−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
𝑦 = 𝑥 − 2𝑥
𝑥
0
1
−1
2
3
𝑦
0
−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 𝑦 = 𝑎𝑥 + 𝑏 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
𝑦 = 3𝑥 + 1
𝑥 𝑦
0 1
1 4
−1 −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 𝑎𝑥 + 𝑏𝑦 = 𝑐 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
2𝑥 − 3𝑦 = 12
𝑥
0
6
3
𝑦
−4
0
−2
For 𝑥 = 0
2 0 − 3𝑦 = 12
0 − 3𝑦 = 12
−3𝑦 = 12
1
𝑦 = − 12
3
𝑦 = −4
8
For 𝑦 = 0
2𝑥 − 3 0 = 12
2𝑥 − 0 = 12
2𝑥 = 12
1
𝑥 = 12
2
𝑥=6
7
6
5
4
3
2
1
8
7
6
5
4
3
2
1
1
1
2
For 𝑥 = 3
2 3 − 3𝑦 = 12
6 − 3𝑦 = 12
−3𝑦 = 12 + −6
−3𝑦 = 6
1
𝑦=− 6
3
𝑦 = −2
3
4
5
6
7
8
2
3
4
5
6
7
8