We know that if there is a common difference between the y-values as the x-values change 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-values remains the same?
Consider the following table of values:
x | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|
y | 4 | 4 | 4 | 4 | 4 |
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.
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.
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.
Use the following applet to explore horizontal lines:
If a line is horizontal, the points on the line have different x-coordinates but the same y-coordinates.
What is the equation of this line?
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.
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.
Consider the following table of values:
x | 4 | 4 | 4 | 4 | 4 |
---|---|---|---|---|---|
y | 1 | 2 | 3 | 4 | 5 |
x | 4 | 4 | 4 | 4 | 4 |
---|---|---|---|---|---|
y | 1 | 5 | -8 | 13 | 50 |
In this case the gradient is considered to be undefined.
A vertical line has an undefined gradient, and an equation of the form: x=c where c is the x-intercept of the line.
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.
Use the following applet to explore vertical lines:
If a line is vertical, the points on the line have different y-coordinates but the same x-coordinates.
Is the graph of y=2 a horizontal or vertical line?
Consider the points in the plane below.
Which of the following statements is true?
What is the equation of the line passing through these points?
What is the equation of the line that is parallel to the y-axis and passes through the point (-8,3)?
A vertical line has an undefined gradient, and an equation of the form: x=c where c is the x-intercept of the line.
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.