13.12 Horizontal and vertical lines
We know that if there is a common difference between the $$y-values as the $$x-value changes by a constant amount, then there is a linear relationship. But what if there is no change in the $$y-values at all? Or if the $$y-values change but the $$x-value remains the same?
Exploration
Consider the following table of values
| $$x | $$1 | $$2 | $$3 | $$4 | $$5 |
|---|---|---|---|---|---|
| $$y | $$4 | $$4 | $$4 | $$4 | $$4 |
We can see that as the $$x-value increases by $$1, the $$y-value does not change at all. We can think of this as increasing, or decreasing for that matter, by $$0 each step.
We know that in a linear equation of the form $$y=mx+c, $$m is equal to the gradient which is the change in the $$y-value for every increase in the $$x-value by $$1. This means we have a value of $$m=0. That is, the gradient of the line is $$0.
If we extended the table of values one place to the left, i.e. when $$x=0, we would find that $$y still has a value of $$4, this means we have a $$y-intercept of $$4. This means we have a value of $$c=4.
Putting it all together we end up at the equation $$y=0x+4 which simplifies to $$y=4.
But what if the values for $$x and $$y were reversed?
Consider the following table of values
| $$x | $$4 | $$4 | $$4 | $$4 | $$4 |
|---|---|---|---|---|---|
| $$y | $$1 | $$2 | $$3 | $$4 | $$5 |
We can see, in this case, that the $$x-value is not actually changing, and the $$y-value is increasing by $$1 each time. Whatever the $$y-value is, $$x is always equal to $$4, so the equation for this table of values is simply $$x=4.
It doesn't actually matter what the increase in $$y-value is in this case - the table could be as follows, and it would still have the same equation $$x=4.
| $$x | $$4 | $$4 | $$4 | $$4 | $$4 |
|---|---|---|---|---|---|
| $$y | $$1 | $$5 | $$−8 | $$13 | $$50 |
In this case the gradient is considered to be undefined.
A horizontal line has a gradient of zero ($$m=0), and an equation of the form: $$y=c where $$c is the $$y-intercept of the line.
A vertical line has an undefined gradient, and an equation of the form: $$x=c where $$c is the $$x-intercept of the line.
Here are two examples of horizontal lines:
Horizontal lines $$y=2 and $$y=−3
Here are two examples of vertical lines:
Vertical lines $$x=−1 and $$x=4
The $$x-axis is a horizontal line, and every point on it has a $$y-value of $$0 so the equation of the $$x-axis is $$y=0.
The $$y-axis is a vertical line, and every point on it has an $$x-value of $$0 so the equation of the $$y-axis is $$x=0.
Practice questions
Question 1
What is the graph of $$y=2?
A horizontal line
AA vertical line
B
Question 2
Consider the points in the plane below.
Which of the following statements is true?
The set of points lie on a vertical line.
AThe set of points lie on a decreasing line.
BThe set of points lie on an increasing line.
CThe set of points lie on a horizontal line.
DWhat is the equation of the line passing through these points?
$$x=−6
A$$y=x−6
B$$y=−6
C
Question 3
What is the equation of this line?
A Cartesian plane has both x- and y-axes that ranges from $$−10 to $$10. Both axes are marked with major intervals of $$2. A line is plotted on the Cartesian plane. The line passes through the points $$(−4,4) and $$(2,4). The points are not marked and not explicitly labeled.
Question 4
What is the equation of the line that is parallel to the $$y-axis and passes through the point $$(−8,3)?