What is a Linear Function?

A linear function is a mathematical expression that describes a relationship between two variables where one variable is linearly dependent on the other. The general form of a linear function is f(x) = mx + b, where 'm' represents the slope of the line, and 'b' represents the initial value or the y-intercept. The graph of a linear function is always a straight line, which means that a change in one variable causes a uniform change in the other.

Linear functions are a fundamental part of mathematics, often used to describe simple, linear relationships in various contexts. This form of the equation shows that the y-coordinate is linearly dependent on the x-coordinate.

GRAPH OF A LINEAR FUNCTION

The graph of a linear function is a straight line, which can be drawn if we know two points on it. Let's see how to do this:

FIND THE Y-INTERCEPT

The point on the graph where the line intersects the y-axis is (0, b). The point (0, b) tells us the value of y when x is equal to zero.

USE THE SLOPE

The slope 'm' tells us how steep the line is. If 'm' is positive, the line rises from left to right; if 'm' is negative, it falls.

STEPS FOR DRAWING THE GRAPH OF A LINEAR FUNCTION:

1. CALCULATE THE Y-INTERCEPT (b): When x = 0, y = b. This is one point on the line.
2. CALCULATE A SECOND POINT USING THE SLOPE (m): From the point (0, b), move one unit to the right (x + 1) and then 'm' units up or down, depending on the sign of the slope.
3. DRAW THE LINE: Connect both points with a straight line that extends in both directions.

KEY PROPERTIES:

SLOPE (m)

This determines the steepness of the line. If m > 0, the line rises. If m < 0, the line falls. If m = 0, the line is horizontal.

Y-INTERCEPT (b)

This determines where the line intersects the y-axis. It is the value of the function when x = 0.

STRAIGHT LINE

The graph of a linear function is always a straight line.

CONCLUSION

Linear functions are an essential part of mathematics, enabling the understanding of many basic and advanced concepts. Their simplicity and versatility allow for wide application in various scientific and everyday situations.