Relations
A relation in mathematics shows how one quantity is connected to another quantity. A relation can be represented using words, a table of values, an equation, ordered pairs, or a graph.
In most relations, we work with two variables: an independent variable and a dependent variable.
The independent variable is usually represented by x. It is the value that we choose or control. It is normally shown on the horizontal axis of a graph.
The dependent variable is usually represented by y. Its value changes depending on the value of x. It is normally shown on the vertical axis.
For example, suppose a taxi charges a starting fee of 4 dollars and then 2 dollars for every kilometre travelled.
The relation can be written as:
y = 2x + 4
Here, x represents the distance travelled and y represents the total cost. The total cost depends on the distance travelled.
A relation can be linear or non-linear.
A linear relation has a constant rate of change. This means that when x increases by the same amount, y changes by the same amount each time. The graph of a linear relation is a straight line.
For example:
y = 3x + 2
If x = 0, then y = 2.
If x = 1, then y = 5.
If x = 2, then y = 8.
If x = 3, then y = 11.
Every time x increases by 1, y increases by 3. Therefore, the rate of change is 3.
The equation of a linear relation is often written as:
y = mx + b
Here, m represents the slope, or rate of change, and b represents the y-intercept, or initial value.
Another linear example is:
y = 5x
This could represent the cost of buying notebooks for 5 dollars each. If x is the number of notebooks, y is the total cost.
A non-linear relation does not have a constant rate of change. Its graph is not a straight line.
For example:
y = x squared
If x = 0, y = 0.
If x = 1, y = 1.
If x = 2, y = 4.
If x = 3, y = 9.
The changes in y are not constant. The graph is a parabola with its vertex at the origin, (0, 0).
Another non-linear example is:
y = 2 to the power of x
If x = 0, y = 1.
If x = 1, y = 2.
If x = 2, y = 4.
If x = 3, y = 8.
If x = 4, y = 16.
Here, every time x increases by 1, the value of y doubles. This is called an exponential relation.
Relations can also describe real-life situations. For example, the amount of water in a tank can depend on time, the distance travelled by a car can depend on time, and the total price of an item can depend on the number purchased.
In summary, a relation shows a connection between two variables. The independent variable is the input, and the dependent variable is the output. Linear relations have a constant rate of change and form straight lines, while non-linear relations have a changing rate of change and form curves or other shapes.
