Unit 4 – Graphs, Linear Equations and Functions
Unit 4.1 – The Rectangular Coordinate System
Plotting Ordered Pairs
Ordered pair – a pair of numbers in parentheses in which the order of the numbers is important.
Graph by using two perpendicular number lines that intersect at 0. The common 0 point is called
the origin.
The horizontal line is referred to as the x-axis and the vertical as the y-axis.
In an ordered pair the first number is the position relative to the x-axis and the second is the
position relative to the y-axis.
The x-axis and y-axis make up a rectangular or Cartesian (after Rene Descartes) coordinate
system.
To locate or plot a point, we move to along the x-axis to the point corresponding to the first
number in the ordered pair and from there move parallel to the y-axis to the second number in
the ordered pair.
Careful: the notation for ordered pairs and open intervals is the same.
4 quadrants of the coordinate system. Where are they?
Find ordered pairs that satisfy a given equation.
An ordered pair is a solution to a given equation if that pair makes the equation true.
Ex. (6, -2) is a solution for 2 x  3 y  6 because 26  3 2  12  6  6 . By the same token,
(5,1 ) is not a solution because 25  31  10  3  13 which is obviously not 6.
Complete the ordered pairs for 2 x  3 y  6 .
(-3, ), ( , -4), now find any other ordered pair that satisfies the equation.
Graph lines.
The graph of an equation is the set of points corresponding to all ordered pairs that satisfy the
equation. It gives a “picture” of the equation. To graph, just plot ordered pairs until you have
enough to suggest the shape of the graph.
The equation 2 x  3 y  6 is called a first-degree equation because it has no term with a variable
to a power greater than one.
The graph of any first-degree equation in two variables is a straight line. Because of this, these
equations are called linear equations in two variables.
The standard form of a linear equation in two variables is Ax  By  C , where A, B, and C are
real numbers and A and B are not both 0.
A straight line can be graphed with only two points. Useful points are the x-intercept (the point
at which the line intersects the x-axis) and the y-intercept (the point at which the line intersects
the y-axis). To find these points, let y = 0 and solve the equation to find the x-intercept and let x
= 0 and solve to find the y-intercept. (Be careful, as not all straight line graphs have an xintercept or a y-intercept.)
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Unit 4 – Graphs, Linear Equations and Functions
Find the x- and y-intercepts of 4 x  y  3 and graph the equation.
Horizontal and Vertical Lines
An equation in the form y  A (A not 0) has no x-intercept and is thus a horizontal line (i.e.
there is no value of x corresponding to y = 0).
An equation in the form x  A (A not 0) has no y-intercept and is thus a vertical line (i.e. there is
no value of y corresponding to x = 0).
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