# 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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