Year 10 Interactive Maths - Second Edition

## Variation

Models of physical situations are often arrived at by recording the values of variables in a table and by sketching graphs. Examining the tabulated values and the graph, we try to define the relationship between the variables by an equation. The equation thus derived is called a mathematical model.

If the value of a dependent quantity changes as a result of changes in the value of an independent quantity, then we say that one quantity varies with respect to the other. However, the type of variation depends on the relationship.

### Direct Proportion (or Variation)

A cyclist's progress over a journey of 120 km is recorded. The results are tabulated below and plotted on a graph.

In such a case, we say that:

This is written as:

From the tabulated values, we find that:

###### In general:

The constant k is called the constant of variation or the constant of proportionality.

Graphically, this relation takes the shape of a straight line with gradient equal to the value of k.

#### Example 9

If y varies directly as x and y = 15 when x = 5, find the formula connecting x and y. Hence find y when x = 8.

##### Solution:

This is the required formula.

#### Example 10

Use the following diagram to find the relationship that exists between x and y.  Hence find y when x is 9.5.