Graph of a Linear Function

INTRODUCTION

A linear function is one of the basic functions in mathematics and has numerous practical applications. The graphical representation of a linear function is a straight line, which has a characteristic constant slope.

BASIC PROPERTIES OF A LINEAR FUNCTION

A linear function is a function that can be written in the form f(x) = mx + b, where:

• 'm' is the slope (gradient) of the line,
• 'b' is the y-intercept (the point where the line crosses the y-axis).

SLOPE OF THE LINE

The slope of the line tells us how steep the line is. If 'm' is positive, the line is increasing; if 'm' is negative, it is decreasing. If 'm' = 0, the line is horizontal and represents a constant function.

Y-INTERCEPT

The point 'b' on the y-axis is the point where the line intersects the y-axis. This value tells us where the line is located in the vertical direction.

EXAMPLE OF DRAWING THE GRAPH OF A LINEAR FUNCTION

Let's take the linear function f(x) = 2x + 3 as an example.

1. DRAW THE COORDINATE SYSTEM: First, draw the x-axis and y-axis.
2. MARK THE Y-INTERCEPT: On the y-axis, mark the point (0, 3), since b = 3.
3. USE THE SLOPE: Since m = 2, this means that for every unit we move to the right (positive direction on the x-axis), we move two units up (positive direction on the y-axis).
4. PLOT A SECOND POINT: Choose any value for x, for example, x = 1, and calculate y = 2*1 + 3 = 5. This gives us a second point (1, 5).
5. CONNECT THE POINTS: Use a ruler to connect the points (0, 3) and (1, 5). The line obtained is the graph of the function f(x) = 2x + 3.