2.13 Connecting descriptions, tables, equations, and graphs of lines

Constructing a table of values from an equation

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

Worked example

Question 1

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

$y=3x-5$y=3x−5

Think: The table of values for this equation connects the $x$x values to the $y$y value that results from substituting that $x$x value into the original equation . Let's complete the table of values below:

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

Do: To substitute $x=1$x=1 into the equation $y=3x-5$y=3x−5, 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	$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

 

Constructing a graph 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	$1$1	$2$2	$3$3	$4$4
$y$y	$-2$−2	$1$1	$4$4	$7$7

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 coordinate 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). We can do the same for the other ordered pairs from the table, resulting in the graph below. (Note that the dotted lines are simply a visual aid, they do not represent part of the graph.)

To complete the graph of the equation $y=3x-5$y=3x−5 we will connect the points that we graphed with a straight line.

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

 

Practice Question

QUESTION 2

 

Constructing a table of values from a graph

We just practiced 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.

 

We're going to practice completing tables of inputs and outputs given a graph.

Worked example

Question 3

Given the following graph, fill in the table.

$x$x	$2$2	$4$4	$6$6	$8$8	$10$10	$12$12	$14$14	$16$16
$y$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	$16$16
$y$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	$16$16
$y$y	$1$1	$2$2	$3$3	$4$4	$5$5	$6$6	$7$7	$8$8

 

Practice questions

Question 4

Question 5

 

Linear representations in the real world

Now that we know how to represent linear functions as equations, tables, and graphs we can put this knowledge to use to solve a variety of real-world problems.

Some examples will be the best way to show you how the mathematics we have learned can be applied to everyday situations.  

Practice questions

Question 6

Question 7