Download Chapter 3: Graphing

Chapter 3: Graphing Linear equations
Previously, we’ve practiced how to solve equations. Today we will look at how to graph these equations so that
we can visualize the relationship between variables.
Consider the following example:

The online retail sales of jewelry y, in billions of dollars is predicted by
y = 0.81x + 2
Where x is the number of years since 2003 (that is, x = 0 corresponds to 2003, x = 5 corresponds to
2008 and so on).
1.
Determine the online retail sales of jewelry in 2003 (in other words, when x = 0)
2.
Determine the online retail sales of jewelry in 2008:
3.
Determine the online retail sales of jewelry in 2015:
4.
Plot the answers from problems 1, 2, and 3, on the coordinate plane.
18
16
14
12
10
8
6
4
2
0
0
5
10
15
5.
Carefully connect the dots that you plotted. This should make a straight line.
6.
Use this line graph to estimate online retail sales in 2010.
7.
Use the line graph to estimate at what year will sales will be about 10 billion
From example 7, we can see that graphing the relation between two variables is often a good way to estimate and
predict solutions to an equation.
Notice that all points on this line that you drew are solutions to the line
y = 0.81x + 2
For example, notice that the coordinate (5, 6.5) is a solution to the equation (as completed in problem 2) and it is
also a coordinate that appears on the line.
Equations where the graphs result in a straight line are called linear equations
Notice from your graph on the previous page that y – 0.81x + 2 is a linear equation .
For each of the following equations,
 find the corresponding y values for the given x values
 plot the coordinates on the graph
 connect the coordinates to determine if it is a straight line
 label each line that you create with the equation
8.
y = 3x – 4
a.
when x = 0, y =
b.
when x = -3, y =
c.
when x = 4, y =
9.
y = 2x + 1
a.
when x = 0, y =
b.
when x = -3, y =
c.
when x = 4, y =
10.
y = 2x
a.
when x = 0, y =
b.
when x = -3, y =
c.
when x = 4, y =
8
6
4
2
0
-8
-6
-4
-2
0
2
4
6
8
-2
-4
-6
-8
Notice that all of these equation fit the general form of a line
y = mx + b
Where m and b are some constants.
y = 3x – 4
y = 2x + 1
y = 2x .
11.
Look back at problems 8, 9, and 10. Consider what happens when you plug in x = 0.
 What are the corresponding y values?

How does this value relate to the “b” part of the y = mx + b form?

Where does this point appear on our graph?
We call the point that the graph intersects the y-axis the y-intercept.
Using this information, we can derive the following definition:
For a linear equation y = mx + b, the “b” value is the y-intercept
Now we need to consider what the “m” constant means.
Graph the following equations by finding the coordinates (0, _), (1, _), and (2, _), plotting these
coordinates, and then connecting the dots on the graph.
12.
y = 1x + 1
13.
y = 2x + 1
14.
y = 3x + 1
8
6
4
2
0
-8
-6
-4
-2
0
-2
-4
-6
-8
2
4
6
8
Graph the following equations by finding the coordinates (0, _), (1, _), and (2, _), plotting these
coordinates, and then connecting the dots.
15.
y = -1x + 1
16.
y = -2x + 1
17.
y = -3x + 1
8
6
4
2
0
-8
-6
-4
-2
0
2
4
6
8
-2
-4
-6
-8
Conclusion for y = mx + b:
18.
If m is positive, the graph slants _________ from left to right.
19.
As m gets larger (more positive), the graph ______________________
20.
If m is negative, the graph slants _________ from left to right
21.
As m gets smaller (more negative), the graph ______________________
So it seems that the “m” value tells us about the direction and amount of slope of the graph.
Slope of a linear equation:
22.
Consider the equation y 




3
x  1 . Find the following ordered pairs:
2
(0, __)
(2, __)
(4, __)
(6, __)
23.
in these ordered pairs, x is increasing by ___________
24.
in these ordered pairs, y is increasing by ___________
25.
How do these numbers relate to the “m” value in the equation y 
26.
Consider the equation y 



3
x  1?
2
5
x  1 . Find the following ordered pairs:
3
(0, __)
(3, __)
(6, __)
27.
in these ordered pairs, x is increasing by ___________
28.
in these ordered pairs, y is decreasing by ___________
29.
How do these numbers relate to the “m” value in the equation y 
5
x  1?
3
In y = mx + b, the “m” value represents the slope of the line.
slope 
rise
run
Another way to say this is that
slope 
change in
change in
3
x  1 , we know:
2
It is linear because it fits the form y = mx + b
the y-intercept of this line on the graph is the coordinate (0,1).
To find another point on the line, all we have to do is move 3 space upward, and two
spaces to the right.
In Conclusion, when we’re given the equation y 



y
x
Graph each of the following equations on the graph below:
2
x4
3
30.
y
31.
3 y  1x  3
32.
y=4
8
6
4
2
0
-8
-6
-4
-2
0
2
4
6
8
-2
-4
-6
-8
33.
Find the equation of the line with a slope of -3 that goes through the coordinate (5, 6).
34.
Find the equation of the line that goes through the points (2, 3) and (5, 8).
35.
When a furniture company priced their sofa at $750, they sold 1,200 in a month. When the
company sold their sofas at $900, they sold 975.
Write the equation representing the cost of the sofas in relation to the price they were sold. (Use
the price as your x variable and the amount sold as the y variable).