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`-value 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 $xy$`x``y`-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.

The word **intercept **in mathematics refers to a point where a line or curve crosses or intersects with the axes.

- We can have $x$
`x`-intercepts: where the line or curve crosses the $x$`x`-axis. - We can have $y$
`y`intercepts: where the line or curve crosses the $y$`y`-axis.

Consider what happens as a point moves up or down along the $y$`y`-axis. It will eventually reach the origin $\left(0,0\right)$(0,0) where $y=0$`y`=0. Now, if the point moves along the $x$`x`-axis in either direction, the $y$`y`-value is still $0$0.

Similarly, consider what happens as a point moves along the $x$`x`-axis. It will eventually reach the origin where $x=0$`x`=0. Now, if the point moves along the $y$`y`-axis in either direction, the $x$`x`-value is still $0$0.

This interactive demonstrates the idea behind the coordinates of $x$`x` and $y$`y`-intercepts.

Intercepts

The $x$`x`-intercept occurs at the point where $y=0$`y`=0.

The $y$`y`-intercept occurs at the point where $x=0$`x`=0.

$x$`x`-intercepts occur when the $y$`y`-value is $0$0. So let $y=0$`y`=0 and then solve for $x$`x`.

$y$`y`-intercepts occur when the $x$`x`-value is $0$0. So let $x=0$`x`=0 and then solve for $y$`y`.

Alternatively we can read the $y$`y`-intercept value from the equation when it is in the form $y=mx+c$`y`=`m``x`+`c`. The value of $c$`c` is the value of the $y$`y`-intercept.

We can also graph a line by identifying the gradient and the $y$`y`-intercept from the equation when it is in the form $y=mx+c$`y`=`m``x`+`c`.

We know that the $y$`y`-intercept occurs at $\left(0,c\right)$(0,`c`), and the gradient is equal to $m$`m`. Using this information we can plot the point at the $y$`y`-intercept (or any other point by substituting in a value for $x$`x` and solving for $y$`y`) and then move right by $1$1, and up (or down if $m$`m` is negative) by $m$`m`.

As as an example, if we have the equation $y=2x+3$`y`=2`x`+3, then we know the $y$`y`-intercept is at $\left(0,3\right)$(0,3) and as the gradient is $2$2, another point will be at $\left(1,3+2\right)=\left(1,5\right)$(1,3+2)=(1,5).

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...

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

What are the coordinates of the $y$

`y`-intercept?Give your answer in the form $\left(a,b\right)$(

`a`,`b`).What are the coordinates of the $x$

`x`-intercept?Give your answer in the form $\left(a,b\right)$(

`a`,`b`).Now, sketch the line $y=2x-2$

`y`=2`x`−2:

Loading Graph...

Sketch the line $y=-x-5$`y`=−`x`−5 using the $y$`y`-intercept and any other point on the line.

- Loading Graph...

Sketch the line that has a gradient of $-3$−3 and an $x$`x`-intercept of $-5$−5.

- Loading Graph...

determines the midpoint, gradient and length of an interval, and graphs linear relationships