Rectangular Coordinates
We begin with two real number lines located in the same plane: one horizontal and one
vertical. The horizontal line is called the x-axis, the vertical line called the y-axis, and
the point of intersection called the origin. We assign coordinates to every point on these
number lines using a convenient scale (usually the same).
Definition: Any point on the plane can be located by using an ordered pair. The notation
(a, b) is the point on the plane whose x-coordinate is a and y-coordinate is b.
y
(a,b)
b
a
(0,0)
x
Note:
• The origin has a value of 0 on both the x-axis and the y-axis
• Points on the x-axis to the right of the origin are associated with positive real numbers
and points to the left of the origin are associated with negative real numbers. Similarly,
points on the y-axis above the origin are associated with positive real numbers and
points below the origin are associated with negative real numbers.
Warning: (a, b) can mean a point or an interval.
Note:
• The graph of y = 0 is the x-axis.
• The graph of x = 0 is the y-axis.
Distance Formula
p
The distance d between points (x, y) and (a, b) is d = (x − a)2 + (y − b)2 .
Example: The distance between (1, 3) and (2, 5) is
p
(1 − 2)2 + (3 − 5)2 =
√
12 + 2 2 =
Example: The distance between points (−2, 1) and (2, 3) is
p
[2 − (−2)]2 + [3 − 1]2 =
√
√
20 = 2 5
Mid-point Formula
For any two points (x1 , y1 ) and (x2 , y2 ), the mid-point M between the two points is:
2 y1 +y2
M = x1 +x
, 2
2
1
√
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Graphs of Equations
Definition: An equation in two variables, say x and y, is a statement in which two expressions involving x and y are equal. Any values of x and y that result in a true statement are
said to satisfy the equation or are called solutions to the equation.
Definition: The graph of an equation in two variables x and y consists of the set of points
in the xy-plane whose coordinates (x, y) satisfy the equation.
Example: Determine if the points A = (2, 3), and B = (2, −2) are on the graph of the
equation 2x − y = 6.
For the point A, check to see if x = 2, y = 3 satisfies the equation:
2(2) − 3 = 4 − 3 = 1 6= 6. So A is not on the graph of the equation.
For the point B, check to see if x = 2, y = −2 satisfies the equation:
2(2) − (−2) = 4 + 2 = 6. So B is on the graph.
Definition: An x-intercept is the x-coordinate of a point where the graph crosses or touches
the x-axis. A y-intercept is the y-coordinate of a point where the graph crosses or touches
the y-axis.
The graph crosses the x-axis when the y-coordinate is 0.
• To find the x-intercepts, set y to 0 and solve for x.
• To find the y-intercepts, set x to 0 and solve for y.
Examples:
1. Find the x and y-intercepts of y − x2 + 4 = 0.
x-intercepts: (set y = 0): 0 − x2 + 4 = 0 ⇐⇒ −x2 = −4 ⇐⇒ x2 = 4
⇐⇒ x = −2, 2. So the x-intercepts are −2, 2.
y-intercepts: (set x = 0): y − 0 + 4 = 0 ⇐⇒ y = −4. So the y-intercept = −4.
2. Find the x and y-intercepts of y − x2 − 4 = 0.
x-intercepts: (set y = 0): 0 − x2 − 4 = 0 ⇐⇒ x2 = −4 So x-intercepts = none.
y-intercepts: (set x = 0): y − 0 − 4 = 0 ⇐⇒ y = 4. So y-intercept = 4.
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Graphing Lines
Definition: The slope of a line is the ratio of the vertical change in y over a horizontal
change in x. We often call the vertical change in y the rise and the horizontal change in x
the run.
Theorem:
• The slope of the line through (x1 , y1 ) and (x2 , y2 ) is m =
y2 −y1
x2 −x1 .
• The equation of the line through (x1 , y1 ) with slope m is y − y1 = m(x − x1 ).
• The equation of the line with slope m and y-intercept b is y = mx + b.
• The slope of a horizontal line is 0.
• The slope of a vertical line is undefined.
• The equation of the vertical line through (a, b) is x = a.
Examples:
1. Find the slope of the line through (2, 3) and (4, 3).
m=
3−3
4−2
= 0.
2. Find the line through (2, 5) and (4, 1).
m=
5−1
2−4
= −2,
y − 5 = −2(x − 2),
y = −2x + 9.
3. Find the equation of the line with slope −8, y-intercept = -6.
y = −8x − 6
Theorem: Two non vertical lines are parallel if and only if their slopes are equal and they
have different y-intercepts.
Theorem:Two non vertical lines are perpendicular if and only if the product of their slopes
is −1.
Example: Find an equation for the line that contains the point (2, −3) and is parallel to
the line 2x + y = 6.
2x + y = 6
⇒
y = −2x + 6
Since we’re looking for a parallel line, the line must have slope −2 and the form
y = −2x + b.
Plugging in the point (2, −3) we can solve for the y-intercept b:
⇒
−3 = −2(2) + b
⇒
b = 1.
So the equation for the line becomes y = −2x + 1.
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