5.02 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. Such relationships are often represented by points connected by lines and curves. The points on the cartesian plane can be generated from a pattern or an equation relating the variables involved. The points are often presented in a table of values. For example, a table of values might look like the following:

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

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 represented.

 

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 by 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).

Number of triangles ($x$x)	$1$1	$2$2	$3$3	$4$4
Number of matches ($y$y)	$3$3	$5$5	$7$7	$9$9

 

Practice question

Question 1

 

Constructing a table of values 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 result from substituting in a variety of $x$x-values. Let's complete the table of values below:

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

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:

$y$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	$1$1	$2$2	$3$3	$4$4
$y$y	$-2$−2

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:

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

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

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

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:

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:

 

Constructing a table of values from a graph

We just practised graphing a table of values. Now we will go backwards and complete a table of values from an already constructed graph. Remember that every graph has an $x$x-axis and a $y$y-axis.

The numbers on the $x$x-axis represent the independent variable and are sometimes called the inputs, while the numbers on the $y$y-axis represent the dependent variable and are called the outputs.

 

Worked example

Example 1

Given the following graph, fill in the table.

$x$x	$2$2	$4$4	$6$6	$8$8	$10$10	$12$12	$14$14
$y$y

Think: Notice on the graph that when the input ($x$x) is $2$2, the output ($y$y) is $1$1. This corresponds with the ordered pair $\left(2,1\right)$(2,1) on the line.

Do: We can fill this output in the table.

$x$x	$2$2	$4$4	$6$6	$8$8	$10$10	$12$12	$14$14
$y$y	$1$1

We can use this method to fill in the entire table as shown below.

$x$x	$2$2	$4$4	$6$6	$8$8	$10$10	$12$12	$14$14
$y$y	$1$1	$2$2	$3$3	$4$4	$5$5	$6$6	$7$7

Practice questions

Question 5

 

Contextual linear relationships in a table of values

Real world situations often involve the collection of data, which is commonly displayed in a table of values. The independent variable (often but not always called $x$x) is conventionally displayed in the first row of the table, followed by the dependent variable (often but not always called $y$y), in the second row.

We may first want to decide whether the values in the table represent a linear relationship. To do this, we could plot the values on a coordinate plane. If we can draw a straight line through all of the points, then we have a linear relationship. We could also check directly from the table by checking if as the $x$x-values increase by a constant amount, the $y$y-values also change by a constant amount.

Once we know that the relationship between the variables is linear, we can work out the gradient and vertical intercept, and express the relationship as a linear equation (or function).

 

Worked example

Example 2

Consider the points in the table. The time ($x$x) is measured in minutes and temperature in degrees Celsius($^\circ$°C).

Time $(x)$(x)	$2$2	$4$4	$6$6	$8$8
Temperature $(y)$(y)	$10$10	$16$16	$22$22	$28$28

(a) Determine the gradient for the linear function represented by the table of values.

Think: The $x$x-values increase by the same amount. We can determine how the $y$y-values change for a given change in $x$x in order to calculate the gradient.

Do:

We see a constant change in $y$y (the rise) of $6$6 for an equivalent change in $x$x (the run) of $2$2. Therefore, we can calculate the gradient as follows:

$m$m	$=$=	$\frac{\text{rise}}{\text{run}}$riserun​
	$=$=	$\frac{6}{2}$62​
	$=$=	$3$3

(b) Interpret the value of the gradient found in part (a).

Think: The gradient gives the rate of change of the dependent variable($y$y) with respect to the independent variable($x$x).

Do: The rate of change of the linear function is $3$3 $^\circ$°C/min. That is, as the time increases by $1$1 minute the temperature increases by $3$3 $^\circ$°C. 

 

(c) Determine the equation of the line, passing through the points given in the table, in gradient-intercept form.

Think: From part (a) we have that the relationship between $x$x and $y$y is of the form: $y=3x+c$y=3x+c. For the first value in the table, when $x=2$x=2, $y=10$y=10, what would we have to add to $3\times2$3×2 to obtain $10$10?

Do: $6+4=10$6+4=10, so $c=4$c=4. Hence, the linear rule that fits the table of values is $y=3x+4$y=3x+4.

Reflect: Check the rule matches the other sets of points in the table. Can you also see the $y$y-intercept would be $4$4 by continuing the pattern in the table to the left?

 

Practice questions

question 6

question 7