Graphs of sloping lines through the origin
Categories: gcse graphs
In this section, we will look at the graphs of sloping straight lines that go through the origin.
These lines all have the equation:
$$ y = a x $$
but use different values for a.
The line y = 2x
This graph shows a graph of the straight line y = 2x:
We can calculate points that are on the line, using the formula. We simply chose an x-value (for example 2), and multiply it by 2 to get the y-value (which would be 4). Here are some other points that are marked on the graph:
- (-1, -2)
- (1, 2)
- (2, 4)
The dashed yellow line has the equation y = x. We can see that the line y = 2x is steeper than the line y = x.
The line y = ⅓x
This graph shows a graph of the straight line y = ⅓x:
Again, we can calculate points that are on the line. We simply chose an x-value (for example 3), and multiply it by one third.
Multiplying by one third is the same as dividing by 3, so this would give a y-value of 1. Here are some other points that are marked on the graph:
- (3, 1)
- (6, 2)
We can see that this time the line is less steep than the line y = x.
Using negative values
We can use a negative multiplier, for example, y = -3x. In this case, if we take an x-value of 1, we get a y-value of -3. Similarly, for an x-value of -1, we get a y-value of 3.
Here is the graph:
This line slopes downwards. Compared to the dashed yellow line y = -x, the line y = -3x is steeper but in the negative direction.
Here is a graph of the equation y = -½x:
This graph goes through the points (-4. 2) and (4, -2), calculated in the same way as before. This time the curve is less steep than y = -x.
Rules
- A straight line passing through the origin has the equation y = ax.
- If a is positive the line slopes upwards.
- If a is negative the line slopes downwards.
- The greater the absolute value of a, the steeper the slope.
See also
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