5.09 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.
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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.)
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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.
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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
Consider the equation $y=3x+1$y=3x+1.
Complete the table of values below:
$x$x $-1$−1 $0$0 $1$1 $2$2 $y$y $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Plot the points in the table of values.
Loading Graph...Draw the graph of $y=3x+1$y=3x+1.
Loading Graph...
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
Given the following graph, fill in the table.
$x$x $1$1 $2$2 $3$3 $4$4 $y$y $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
Question 5
Buzz recorded his savings (in $dollars$dollars) over a few months in the graph given.
Complete the table.
Months $1$1 $2$2 $3$3 $4$4 Savings $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Buzz estimates that he will have exactly $\$60$$60 in his savings by month $5$5. Is this true or false?
True
AFalse
B
Linear representations in the real world
Now that we know how to represent linear functions as equations, tables, and graphs we can put this knowledge to use to solve a variety of real-world problems.
Some examples will be the best way to show you how the mathematics we have learned can be applied to everyday situations.
Practice questions
Question 6
A diver starts at the surface of the water and begins to descend below the surface at a constant rate. The table shows the depth of the diver over $5$5 minutes.
| Number of minutes passed ($x$x) | $0$0 | $1$1 | $2$2 | $3$3 | $4$4 |
|---|---|---|---|---|---|
| Depth of diver in meters ($y$y) | $0$0 | $1.4$1.4 | $2.8$2.8 | $4.2$4.2 | $5.6$5.6 |
What is the increase in depth each minute?
Write an equation for the relationship between the number of minutes passed ($x$x) and the depth ($y$y) of the diver.
Enter each line of work as an equation.
In the equation, $y=1.4x$y=1.4x, what does $1.4$1.4 represent?
The change in depth per minute.
AThe diver’s depth below the surface.
BThe number of minutes passed.
CAt what depth would the diver be after $6$6 minutes?
We want to know how long the diver takes to reach $12.6$12.6 meters beneath the surface.
If we substitute $y=12.6$y=12.6 into the equation in part (b) we get $12.6=1.4x$12.6=1.4x.
Solve this equation for $x$x to find the time it takes.
Question 7
A carpenter charges a callout fee of $\$150$$150 plus $\$45$$45 per hour.
Write an equation to represent the total amount charged, $y$y, by the carpenter as a function of the number of hours worked, $x$x.
What is the slope of the function?
What does this slope represent?
The total amount charged increases by $\$45$$45 for each additional hour of work.
AThe minimum amount charged by the carpenter.
BThe total amount charged increases by $\$1$$1 for each additional $45$45 hours of work.
CThe total amount charged for $0$0 hours of work.
DWhat is the value of the $y$y-intercept?
What does this $y$y-intercept represent?
Select all that apply.
The total amount charged increases by $\$150$$150 for each additional hour of work.
AThe maximum amount charged by the carpenter.
BThe callout fee.
CThe minimum amount charged by the carpenter.
DFind the total amount charged by the carpenter for $6$6 hours of work.