4.02 Tables of values

As mentioned in the previous lesson, the cartesian plane is used to locate points and graphically represent mathematical relationships between two variables. Mathematical relationships are represented by points connected by lines and curves. The points on the cartesian plane can be generated from a pattern or an equation. The points are often presented in a table of values. For example, a table of values might look like the following:

Each column of $x$x- and $y$y-values represents a pair of coordinates, or an ordered pair. From this table, for example, $\left(3,10\right),\left(6,19\right),\left(9,28\right),\left(12,37\right)$(3,10),(6,19),(9,28),(12,37) are the coordinate pairs.

Constructing a table of values

From a pattern

Let's consider the pattern below. The pattern starts with a triangle made out of matchsticks and continues adding two additional matchsticks to each subsequent iteration of the pattern.

The table of values for this pattern connects the number of triangles made ($x$x), with the number of matches needed ($y$y).

Practice question

Question 1

From an equation

We can also construct a table of values using an equation:

$y=3x-5$y=3x−5

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

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

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

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

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

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

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

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

Each column in a table of values may be grouped together in the form $\left(x,y\right)$(x,y). The table of values has the following ordered pairs:

Practice questions

Question 2

Consider the equation $y=7x$y=7x.

1. What is the value of $y$y when $x=-5$x=−5?

2. What is the value of $y$y when $x=0$x=0?

3. What is the value of $y$y when $x=5$x=5?

4. What is the value of $y$y when $x=10$x=10?

5. Complete the table of values below:

   $x$x	$-5$−5	$0$0	$5$5	$10$10
   $y$y	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$

Reveal Solution

Question 3

Consider the equation $y=5x+6$y=5x+6.

1. Complete the table of values below:

   $x$x	$-10$−10	$-5$−5	$0$0	$5$5
   $y$y	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$

Reveal SolutionWatch video

 

Practical applications

Many practical applications give rise to mathematical relationships. Creating a table of values that represents these applications is a good starting point for further exploring the nature of the relationship.

Practice question

Question 4

Drawing a straight line from a table of values

We can plot each ordered pair in our table of values as a point on the cartesian plane. Consider the following table of values mapped to its respective ordered pairs:

We can plot the ordered pair $\left(a,b\right)$(a,b) by first identifying $x=a$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 imagine a vertical line through this point. Then we identify $y=4$y=4 along the $y$y-axis and imagine 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).

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

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=3x−5 which we used to complete the table of values. From this, we can gain a clear picture of the mathematical relationship represented by the equation.

 

Practice question

Question 5

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

1. Complete the table of values below:

   $x$x	$-1$−1	$0$0	$1$1	$2$2
   $y$y	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$

2. Plot the points in the table of values.

   Loading Graph...

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

   Loading Graph...

Reveal SolutionWatch video

 

Drawing a straight line from intercepts

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.

For example, consider the equation:

$y=3x-6$y=3x−6

And the table of values:

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

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

• An $x$x-intercept has the form $\left(a,0\right)$(a,0) and is a point that lies on the $x$x-axis.
• A $y$y-intercept has the form $\left(0,b\right)$(0,b) and 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=3x−6 which passes through these two points (or any other pair of points):

 

Finding $x$x and $y$y intercepts

To find the $x$x-intercept put $y=0$y=0

To find the $y$y-intercept put $x=0$x=0

 

Practice questions

QUESTION 6

Consider the equation $y=-4x+4$y=−4x+4.

1. Find the coordinates of the $y$y-intercept.

2. Find the coordinates of the $x$x-intercept.

3. Sketch a graph of the line below.

   Loading Graph...

Reveal SolutionWatch video

QUESTION 7

Consider the line with equation: $3x+y+2=0$3x+y+2=0

1. Solve for the $x$x-value of the $x$x-intercept of the line.

2. Solve for the $y$y-value of the $y$y-intercept of the line.

3. Plot the line.

   Loading Graph...

Reveal SolutionWatch video