In Further Mathematics, the graphs dealt with are in the form y = kx^n where *k* and *n* are constants. The *k* value does not affect the shape of the graph but rather dilates (stretches or compresses) the graph. Changes in *n* change the shape of the graph drastically. Here we will explore different non-linear graphs by varying the value *n*.

## Variations in *n* value

### n = 1, y = kx

These are straight lines through the origin. These are used in linear models.

### n = -1, y = k / x

These are called **hyperbolas**. Here, with the increase in *x*, the graph *approaches y *= 0 but never reaches it. Likewise, as *y* goes to infinity, *x* *approaches *0. The graph never crosses the *y* or *x* axis. These lines are called **asymptotes.**

### n = 2, y = kx^2

These graphs are called **parabolas. **The point where the graph temporarily has no gradient at the origin (0,0) is called the **vertex** or turning point.** **These graphs are symmetrical about the *y*-axis

### n = -2, y = k / x^2

When *n* = -2, the graph produced is called a **truncus. **A truncus also has asymptotes.

### n = 3, y = kx^3

This is a **cubic **graph where *n* = 3. Varying *k* values dilates the graph, and a negative *k* value reflects it.

## Linear representation of non-linear relations

By changing the label on the axis, it is possible to draw a linear graph with non-linear relations.

### Example

Consider the relationship y = 0.5x^3. The graph of this would appear as the following (left side):

If the values for *x* were transformed into x^3 and graphed against the y values, the points would resemble a linear relationship. Since the data was transformed it must be noted that the axis has also changed.

By looking at the right-side graph, the gradient can be observed to be 0.5. Usually a linear graph which crosses the origin with a gradient of 0.5 would have the equation y = 0.5x. However, since the independent variable (x-axis) is x^3, the equation would actually be y = 0.5x^3.

**See also:**

## Feedback

Want to suggest an edit? Have some questions? General comments? Let us know how we can make this resource more useful to you.