Hong Kong
Stage 1 - Stage 3

# Graphing Linear Equations from Ordered Pairs

Lesson

## Constructing a table of values

Recall that a table of values is what we call a table that shows the values of two quantities (usually represented by $x$x and $y$y) that are related in some way. As an example, a table of values might look like:

 $x$x $y$y $3$3 $6$6 $9$9 $12$12 $10$10 $19$19 $28$28 $37$37

#### Exploration

Let's construct our own table of values using the following equation:

$y=3x-5$y=3x5

The table of values for this equation connects the $y$y-value that result from substituting in a variety of $x$x-values. Let's complete the table of values below:

 $x$x $y$y $1$1 $2$2 $3$3 $4$4

To substitute $x=1$x=1 into the equation $y=3x-5$y=3x5, we want replace all accounts of $x$x with $1$1.

So for $x=1$x=1, we have that:

 $y$y $=$= $3\left(1\right)-5$3(1)−5 $=$= $3-5$3−5 $=$= $-2$−2

So we know that $-2$2 must go in the first entry in the row of $y$y-values.

 $x$x $y$y $1$1 $2$2 $3$3 $4$4 $-2$−2

Next let's substitute $x=2$x=2 into the equation $y=3x-5$y=3x5.

For $x=2$x=2, we have that:

 $y$y $=$= $3\left(2\right)-5$3(2)−5 $=$= $6-5$6−5 $=$= $1$1

So we know that $1$1 must go in the second entry in the row of $y$y-values.

 $x$x $y$y $1$1 $2$2 $3$3 $4$4 $-2$−2 $1$1

If we substitute the remaining values of $x$x, we find that the completed table of values is:

 $x$x $y$y $1$1 $2$2 $3$3 $4$4 $-2$−2 $1$1 $4$4 $7$7

## Plotting points from a table of values

Each column in a table of values may be grouped together in the form $\left(x,y\right)$(x,y). We call this pairing an ordered pair. Let's return to our table of values:

 $x$x $y$y $1$1 $2$2 $3$3 $4$4 $-2$−2 $1$1 $4$4 $7$7

#### Exploration

The table of values has the following ordered pairs:

$\left(1,-2\right),\left(2,1\right),\left(3,4\right),\left(4,7\right)$(1,2),(2,1),(3,4),(4,7)

We can plot each ordered pair as a point on the $xy$xy-plane.

We can plot the ordered pair $\left(a,b\right)$(a,b) by first identifying where $x=a$x=a along the $x$x-axis and $y=b$y=b along the $y$y-axis.

Take $\left(3,4\right)$(3,4) as an example. We first identify $x=3$x=3 along the $x$x-axis and draw a vertical line through this point. Then we identify $y=4$y=4 along the $y$y-axis and draw a horizontal line through that point. Finally we plot a point where two lines meet, and this represents the ordered pair $\left(3,4\right)$(3,4).

## Drawing a straight line from points on the plane

Now that we have drawn the ordered pairs from the table of values, we can draw the graph that passes through these points.

#### Exploration

In the example above, the line that passes through these points is given by:

This straight line is the graph of $y=3x-5$y=3x5 which we used to complete the table of values.

## Drawing a straight line from a table of values

To draw a line from a table of values, it is useful to plot the significant points and draw the line that passes through them.

#### Exploration

For example, consider the equation:

$y=3x-6$y=3x6

And the table of values:

 $x$x $y$y $0$0 $1$1 $2$2 $3$3 $-6$−6 $-3$−3 $0$0 $3$3

There are two significant ordered pairs, namely the $x$x-intercept and the $y$y-intercept.

• The $x$x-intercept has the form $\left(a,0\right)$(a,0) which is a point that lies on the $x$x-axis.
• The $y$y-intercept has the form $\left(0,b\right)$(0,b) which is a point that lies on the $y$y-axis.

The $x$x-intercept in our example is $\left(2,0\right)$(2,0) and the $y$y-intercept is $\left(0,-6\right)$(0,6).

We can draw the line of $y=3x-6$y=3x6 which passes through these two points.

#### Practice Questions

##### Question 1

Consider the equation $y=4x$y=4x. A table of values is given below.

 $x$x $y$y $-2$−2 $-1$−1 $0$0 $1$1 $-8$−8 $-4$−4 $0$0 $4$4
1. Plot the points in the table of values.

2. Is the graph of $y=4x$y=4x linear?

Yes

A

No

B

##### QUESTION 2

Consider the equation $y=3x+1$y=3x+1.

1. Complete the table of values below:

 $x$x $y$y $-1$−1 $0$0 $1$1 $2$2 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
2. Plot the points in the table of values.

3. Draw the graph of $y=3x+1$y=3x+1.

##### Question 3

Consider the equation $y=-\frac{x}{7}$y=x7.

1. Complete the table of values below:

 $x$x $y$y $-7$−7 $-4$−4 $-3$−3 $0$0 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
2. Draw the graph of $y=-\frac{x}{7}$y=x7.