Download Section 2.1 – Points, Lines and Graphs In this section, we look at the

Section 2.1 – Points, Lines and Graphs
In this section, we look at the Cartesian coordinate system. There are a lot of thing we can do with this: the most
important is to graph equations of lines. Let’s look at the actual graph first.
§1 The Cartesian Coordinate System
The graph is separated into four quadrants. The following table shows the sign of the x- and y- coordinates in each
quadrant.
Quadrant
x-coordinate
y-coordinate
I
+
+
II
+
III
IV
+
So if a point is in quadrant I, it’s x-coordinate is positive and its y-coordinate is positive. Note that if a point lies on the
x-axis, its y-coordinate is 0. Similarly, if a point lies on the y-axis, its x-coordinate is 0.
PRACTICE
1) Plot the following points: (3,5), (-2,3), (-1,-5), (4,-2), (0,-4), and (5,0)
§2 Linear Equations in two variables
Here we start dealing with equations in two variables, such as 2 x  y  5 . A solution to this equation is an ordered
pair (x,y) that satisfies the equation. For example, (3,1) is a solution to the above equation, but (2,1) is not. There are
infinitely many solutions to a linear equation. Please remember that solutions are given in ordered pair form. To find
solutions, you can randomly select values for one of the variables, substitute that value into the equation, and find
the corresponding value for the other variable. Note that the best values to pick are 0,1 and 2!
PRACTICE
2) Is the ordered pair (2,-2) a solution to 3x  2 y  10?
3) For the equation 3x  2 y  10, if y  2 find the corresponding value for x.
4) Find three solutions to the equation y  2 x  3
§3 Lines
The main thing to remember is, if we plot the ordered pairs that are solutions to a particular equation, the picture
that we get is a line!
Look at the points you found for practice problem 4. Plot the points below and connect the points. What shape do
you get?
Hence, we can always graph a linear equation by first plotting points! We only need two but sometimes you may be
asked for three or more.
PRACTICE
5) Graph the linear equation 2 x  y  4
6) Graph the linear equation y  3x  1
§4 Intercepts
Remember earlier we looked at points that lie on the x-axis. It turns out that the corresponding y-value is 0. Similarly,
if a point lies on the y-axis, its x-coordinate is 0. These two points are called intercepts. So given a linear equation, we
always find the intercepts- to find the y-intercept, set x = 0 and solve for y. To find the x-intercept, set y = 0 and solve
for x!
PRACTICE
7) Find the intercepts of 2 x  3 y  6
8) Find the intercepts of y  2 x  1
§5 Slope of a line
The slope of a line represents the steepness. We call it ‘rise over run’, because it tells us the ratio of the vertical
change to its horizontal change. We can always find the slope if we have two points,  x1 , y1  and  x2 , y2  . The
equation is slope = m =
y2  y1
. Make sure you don’t mix up the points and that the y-values go in the numerator!
x2  x1
The slope of the line that passes through (4,3) and 1, 2  is then m 
2  3 5 5


1  4 3 3
PRACTICE
9) Find the slope of the line that passes through the points  3, 2  and 1,1
10) Find the slope of the line that passes through the points  3, 2  and 1,5
Note that slope can be positive or negative. If the slope is negative, the line goes up and to the right (or down and to
the left). If the slope is negative, the line goes up and to the left (or down and to the right).
Slope can also be zero. Horizontal lines have zero slope. For a horizontal line, all the y-values are the same. An
equation such as y  3 is a horizontal line. A line with no slope is a vertical line. In a vertical line, all the x-values are
the same. An equation like x  1 is a vertical line.
PRACTICE
11) Find the slope of the line that passes through the points  3, 2  and  6, 2 
12) Find the slope of the line that passes through the points  3, 2  and  3, 4 
§6 Grade
A slope given in percent form is called a grade. You may have seen these on roads that are steep. For example, an 18%
grade means the road rises 18 feet for EVERY 100 feet (percent literally means ‘per 100’).