# Graphs of sloping lines through the origin

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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