Middle Years

Lesson

A table of values, created using an equation, forms a set of points that can be plotted on a number plane. A line, drawn through the points, becomes the graph of the equation.

We'll begin by creating a table of values for the following equation:

$y=3x-5$`y`=3`x`−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`=3`x`−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 |

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.

Plotting points on a number plane

To plot a point, $\left(a,b\right)$(`a`,`b`), on a number 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`=3`x`−5.

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.

Did you know?

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.

Consider the equation $y=2x$`y`=2`x`.

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

Consider the equation $y=4x+5$`y`=4`x`+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

Consider the equation $y=-2x+4$`y`=−2`x`+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...