We know how to identify if a table of values represents a linear equation, and now we will look at how to display the same information on a Cartesian plane.
Creating a table of values
A table of values, created using an equation, forms a set of points that can be plotted on a Cartesian plane. A line, drawn through the points, becomes the graph of the equation.
Exploration
We'll begin by creating a table of values for the following equation:
$y=3x-5$y=3x−5
The first row of the table will contain values for the independent variable (in this case, $x$x). The choice of $x$x-values is often determined by the context, but in many cases they will be given. To find the corresponding $y$y-value, we substitute each $x$x-value into the equation $y=3x-5$y=3x−5.
| $x$x | $1$1 | $2$2 | $3$3 | $4$4 |
|---|---|---|---|---|
| $y$y |
Substituting $x=1$x=1:
| $y$y | $=$= | $3\times1-5$3×1−5 |
| $=$= | $3-5$3−5 | |
| $=$= | $-2$−2 |
Substituting the remaining values of $x$x, allows us to complete the table:
| $x$x | $1$1 | $2$2 | $3$3 | $4$4 |
|---|---|---|---|---|
| $y$y | $-2$−2 | $1$1 | $4$4 | $7$7 |
Plotting points from a table of values
The $x$x and $y$y value in each column of the table can be grouped together to form the coordinates of a single point, $\left(x,y\right)$(x,y).
Each point can then be plotted on a coordinate plane.
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To plot a point, $\left(a,b\right)$(a,b), on a Cartesian plane, we first identify where $x=a$x=a lies along the $x$x-axis, and where $y=b$y=b lies along the $y$y axis.
For example, to plot the point $\left(3,4\right)$(3,4), we identify $x=3$x=3 on the $x$x-axis and construct a vertical line through this point. Then we identify $y=4$y=4 on the $y$y-axis and construct a horizontal line through this point. The point where the two lines meet has the coordinates $\left(3,4\right)$(3,4).
If we sketch a straight line through the points, we get the graph of $y=3x-5$y=3x−5.
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Notice that when sketching a straight line through a set of points, the line should not start and end at the points, but continue beyond them, across the entire coordinate plane.
To sketch a straight line graph we actually only need to identify two points!
- When checking if a set of points forms a linear relationship, we can choose any two of the points and draw a straight line through them. If the points form a linear relationship then any two points will result in a straight line passing through all the points.
Practice Questions
Question 1
Consider the equation $y=2x$y=2x.
Fill in the blanks to complete the table of values.
$x$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...Is the relationship linear?
Yes, the relationship is linear.
ANo, the relationship is not linear.
B
QUESTION 2
Consider the equation $y=4x+5$y=4x+5.
Fill in the blanks to complete the table of values.
$x$x $-1$−1 $0$0 $1$1 $2$2 $y$y $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Plot the points that correspond to when $x=-1$x=−1 and when $x=1$x=1:
Loading Graph...Now, sketch the line that passes through these two points:
Loading Graph...Does the point $\left(2,13\right)$(2,13) lie on this line?
Yes
ANo
B
Question 3
Consider the equation $y=-2x+4$y=−2x+4.
Fill in the blanks to complete the table of values.
$x$x $0$0 $1$1 $2$2 $3$3 $y$y $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Plot the points that correspond to when $x=0$x=0 and $y=0$y=0:
Loading Graph...Now, sketch the line that passes through these two points:
Loading Graph...