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

Lesson

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=3x5

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=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$35
  $=$= $-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=3x5.

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

$y$y $=$= $3\left(2\right)-5$3(2)5
  $=$= $6-5$65
  $=$= $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=3x5 we will connect the points that we graphed with a straight line.

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

 

Practice Question

QUESTION 2

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...

 

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.

Loading Graph...

  1. $x$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.

Loading Graph...

  1. Complete the table.

    Months $1$1 $2$2 $3$3 $4$4
    Savings $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
  2. Buzz estimates that he will have exactly $\$60$$60 in his savings by month $5$5. Is this true or false?

    True

    A

    False

    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
  1. What is the increase in depth each minute?

  2. 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.

  3. In the equation, $y=1.4x$y=1.4x, what does $1.4$1.4 represent?

    The change in depth per minute.

    A

    The diver’s depth below the surface.

    B

    The number of minutes passed.

    C
  4. At what depth would the diver be after $6$6 minutes?

  5. 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.

  1. 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.

  2. What is the slope of the function?

  3. What does this slope represent?

    The total amount charged increases by $\$45$$45 for each additional hour of work.

    A

    The minimum amount charged by the carpenter.

    B

    The total amount charged increases by $\$1$$1 for each additional $45$45 hours of work.

    C

    The total amount charged for $0$0 hours of work.

    D
  4. What is the value of the $y$y-intercept?

  5. 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.

    A

    The maximum amount charged by the carpenter.

    B

    The callout fee.

    C

    The minimum amount charged by the carpenter.

    D
  6. Find the total amount charged by the carpenter for $6$6 hours of work.

Outcomes

8.F.4

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

8.F.5

Describe qualitatively the functional relationship between two quantities by analyzing a graph (for example Where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

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