Linear functions

Understanding Linear Functions

  • A linear function is a polynomial function of degree one. Basically, it is a function that describes a straight line when graphed.

  • Linear functions have the general form y = mx + c, where m is the gradient slope and c is the y-intercept.

  • The slope or gradient of the line (m) describes how steep the line is. A positive slope indicates an upward trend, while a negative slope indicates a downward trend.

  • The y-intercept (c) is the point where the line crosses the y-axis.

Forming Linear Functions

  • To create a linear function, two pieces of information are generally needed: the slope of the line and the y-intercept.

  • The equation of a straight line with a slope of m and y-intercept of c is given by y = mx + c.

  • For example, a line with a slope of 2 and crosses the y-axis at 4 has the linear function y = 2x + 4.

Features of Linear Functions

  • Linear functions have a constant rate of change. This means the difference in the y-values for any two points on the line is always the same.

  • When the slope of a linear function is zero, the resulting line is horizontal. Such a line is written as y = c, where c is a constant.

  • Linear functions are continuous and are always defined for all real number x-values. There are no breaks or holes in the graph of a linear function.

Graphing Linear Functions

  • Linear functions can be graphed by plotting the y-intercept first, and then using the slope to find another point on the line.

  • The slope of a line can be found using the formula m = (change in y) / (change in x) or m = (y2 - y1) / (x2 - x1).

  • To draw a line with a positive slope, switch from the y-intercept and moves up and to the right. For a line with a negative slope, switch from the y-intercept and move down and to the right.

Interpreting Linear Functions

  • The equation of a linear function provides important real-world information. The slope can often represent rates, such as speed or growth, while the y-intercept can represent starting values or initial conditions.

  • For example, in the linear function modelling a car’s journey, y = 60x + 0 where y is the distance covered in miles, x is time in hours - the slope, 60, can represent the speed of the car in miles per hour and the y-intercept, 0, represents the distance covered at time 0.