Middle Years

Lesson

We know that if there is a common difference between the $y$`y`-values as the $x$`x`-value changes by a constant amount, then there is a linear relationship. But what if there is no change in the $y$`y`-values at all? Or if the $y$`y`-values change but the $x$`x`-value remains the same?

Consider the following table of values

$x$x |
$1$1 | $2$2 | $3$3 | $4$4 | $5$5 |
---|---|---|---|---|---|

$y$y |
$4$4 | $4$4 | $4$4 | $4$4 | $4$4 |

We can see that as the $x$`x`-value increases by $1$1, the $y$`y`-value does not change at all. We can think of this as increasing, or decreasing for that matter, by $0$0 each step.

We know that in a linear equation of the form $y=mx+c$`y`=`m``x`+`c`, $m$`m` is equal to the gradient which is the change in the $y$`y`-value for every increase in the $x$`x`-value by $1$1. This means we have a value of $m=0$`m`=0. That is, the gradient of the line is $0$0.

If we extended the table of values one place to the left, i.e. when $x=0$`x`=0, we would find that $y$`y` still has a value of $4$4, this means we have a $y$`y`-intercept of $4$4. This means we have a value of $c=4$`c`=4.

Putting it all together we end up at the equation $y=0x+4$`y`=0`x`+4 which simplifies to $y=4$`y`=4.

But what if the values for $x$`x` and $y$`y` were reversed?

Consider the following table of values

$x$x |
$4$4 | $4$4 | $4$4 | $4$4 | $4$4 |
---|---|---|---|---|---|

$y$y |
$1$1 | $2$2 | $3$3 | $4$4 | $5$5 |

We can see, in this case, that the $x$`x`-value is not actually changing, and the $y$`y`-value is increasing by $1$1 each time. Whatever the $y$`y`-value is, $x$`x` is always equal to $4$4, so the equation for this table of values is simply $x=4$`x`=4.

It doesn't actually matter what the increase in $y$`y`-value is in this case - the table could be as follows, and it would still have the same equation $x=4$`x`=4.

$x$x |
$4$4 | $4$4 | $4$4 | $4$4 | $4$4 |
---|---|---|---|---|---|

$y$y |
$1$1 | $5$5 | $-8$−8 | $13$13 | $50$50 |

In this case the gradient is considered to be undefined.

Equations of horizontal and vertical lines

A horizontal line has a gradient of zero ($m=0$`m`=0), and an equation of the form: $y=c$`y`=`c` where $c$`c` is the $y$`y`-intercept of the line.

A vertical line has an undefined gradient, and an equation of the form: $x=c$`x`=`c` where $c$`c` is the $x$`x`-intercept of the line.

Here are two examples of horizontal lines:

Here are two examples of vertical lines:

The $x$`x` and $y$`y`-axes

The $x$`x`-axis is a horizontal line, and every point on it has a $y$`y`-value of $0$0 so the equation of the $x$`x`-axis is $y=0$`y`=0.

The $y$`y`-axis is a vertical line, and every point on it has an $x$`x`-value of $0$0 so the equation of the $y$`y`-axis is $x=0$`x`=0.

What is the graph of $y=2$`y`=2?

A horizontal line

AA vertical line

B

Consider the points in the plane below.

Which of the following statements is true?

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The set of points lie on a vertical line.

AThe set of points lie on a decreasing line.

BThe set of points lie on an increasing line.

CThe set of points lie on a horizontal line.

DWhat is the equation of the line passing through these points?

$x=-6$

`x`=−6A$y=x-6$

`y`=`x`−6B$y=-6$

`y`=−6C

What is the equation of this line?

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What is the equation of the line that is parallel to the $y$`y`-axis and passes through the point $\left(-8,3\right)$(−8,3)?