Tables and graphs are useful to find a pattern between the *y*-coordinate and the* x*-coordinate.
Using Tables
We can find a pattern in coordinates by listing their ordered pairs in a
table.
Example 7
Consider the following table.
a. Describe the relationship between the *y*-coordinate and
the *x*-coordinate in words.
b. Find the algebraic relationship between *x* and *y*.
##### Solution:
a. By trial and error, we look for a relationship between the
values of *x* and *y*.
We can describe the relationship between *x *and *y* in words as follows:
The *y*-coordinate is twice the *x*-coordinate
This means *y* is twice *x*.
b. So, the algebraic relationship between the *x*-coordinate
and *y*-coordinate is:
Using Graphs
We can find a pattern in coordinates by drawing a graph of their ordered pairs.
Example 8
a. Plot the following points on a Cartesian plane:
*A*(1, 3), *B*(2, 6), *C*(3, 9), *D*(4, 12), *E*(5,
15), *F*(6, 18)
Use a ruler to join the points *A*, *B*,* C*, *D*, *E* and *F*.
b. Describe the relationship between the *y-*coordinate
and the *x*-coordinate in words.
c. Find the algebraic relationship between *x* and *y*.
##### Solution:
b. We notice that the points lie on a line. Such a pattern is
called a **linear relationship **because it represents a straight line
relationship between the coordinates of the points.
We can describe the relationship between *x *and *y* in words
as follows: The *y*-coordinate is three times the *x*-coordinate.
This means *y* is three times *x*.
c. So, the algebraic relationship between the *x*-coordinate
and *y-*coordinate is:
Example 9
a. Plot the following points on a Cartesian plane:
*A*(2, 0), *B*(3, 1), *C*(4, 2), *D*(5, 3), *E*(6,
4) and *F*(7, 5)
Use a ruler to join the points *A*, *B, C*, *D*, *E* and *F*.
b. Describe the relationship between the *y*-coordinate and the *x*-coordinate
in words.
c. Find the algebraic relationship between *x* and *y*.
##### Solution:
b. We notice that the points lie on a line. Such a pattern is called a **linear
relationship** because it represents a straight line relationship between
the coordinates of the points in the pattern.
We can describe the relationship between *x *and *y* in words
as follows:
The *y*-coordinate is two less than the *x*-coordinate
c. So algebraically, the relationship between the *x*-coordinate and *y*-coordinate is:
Using a Difference Pattern
When we look for a pattern in ordered pairs, we can find the difference
between two successive values of *y*. This allows us to find a rule as
illustrated below.
Consider the following table.
We notice that the values of *x* increase by just one at a time and
the difference between the successive values of *y* is 2. So, the rule
starts off with *y* = 2*x*. Will this give a correct answer from
the table? Let us check.
The answer is no. From the table, when *x* = 1 the
value of *y* should be 5. How do we change our answer from 2 to 5?
We
should add 3.
Check the rule to see if it is correct:
Key Terms
tables, graphs, linear
relationships, difference pattern |