 Middle Years

# 10.04 Horizontal and vertical lines

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?

#### Exploration

Consider the following table of values

 $x$x $y$y $1$1 $2$2 $3$3 $4$4 $5$5 $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=mx+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=0x+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 $y$y $4$4 $4$4 $4$4 $4$4 $4$4 $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 $y$y $4$4 $4$4 $4$4 $4$4 $4$4 $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: Horizontal lines $y=2$y=2 and $y=-3$y=3

Here are two examples of vertical lines: Vertical lines $x=-1$x=1 and $x=4$x=4

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.

#### Practice questions

##### Question 1

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

1. A horizontal line

A

A vertical line

B

A horizontal line

A

A vertical line

B

##### Question 2

Consider the points in the plane below.

Which of the following statements is true?

1. The set of points lie on a vertical line.

A

The set of points lie on a decreasing line.

B

The set of points lie on an increasing line.

C

The set of points lie on a horizontal line.

D

The set of points lie on a vertical line.

A

The set of points lie on a decreasing line.

B

The set of points lie on an increasing line.

C

The set of points lie on a horizontal line.

D
2. What is the equation of the line passing through these points?

$x=-6$x=6

A

$y=x-6$y=x6

B

$y=-6$y=6

C

$x=-6$x=6

A

$y=x-6$y=x6

B

$y=-6$y=6

C

##### Question 3

What is the equation of this line?

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)?