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.

BYes, 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

BYes

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...