10.04 Horizontal and vertical lines
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 | $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=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 | $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$50 |
In this case the gradient is considered to be undefined.
A horizontal line has a gradient of zero ($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-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?
A horizontal line
AA vertical line
B
Question 2
Consider the points in the plane below.
Which of the following statements is true?
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=−6
A$y=x-6$y=x−6
B$y=-6$y=−6
C
Question 3
What is the equation of this line?
Question 4
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)?