Some Common Functions and their Graphs – Constant Functions & Linear Functions

What is a constant function? Constant Functions What is a constant function? A function in the form y = c or f(x) = c, where c is a constant, is called a constant function.

Constant Functions What is a constant function? For example, y = 1 and f(x) = –2 are constant functions.

Consider the constant function y = 1. x –2 2 y 1 2 y 1 ◄ The value of y is always equal to 1, for any value of x. y x (–2, ) 1 (0, ) 1 (2, ) 1 y = 1 y-coordinates of all the points on the line are equal to 1 A straight line parallel to the x-axis

Graph of y = c O y x c 0 O y x c= 0 O y x c 0 (0, c) (0, c) (0, c) The graph of a constant function is a horizontal line passing through (0, c).

Follow-up question The figure shows the graph of a constant function. Write down the algebraic representation of the function. x y 2 1 1 2 3 4 2 4 ∵ The graph is a horizontal line which cuts the y-axis at (0, –3). (0, –3) ∴ The required function is y = –3. y = f(x)

Linear Functions A function in the form y = ax + b or f(x) = ax + b, where a and b are constants and a  0, is called a linear function of x. Example: (i) y = 3x – 1 linear functions of x (ii) f(x) = –2x + 5

The graph cuts the x-axis at P. Graph of y = ax + b O y x y = ax + b The graph cuts the x-axis at P. P

The graph cuts the y-axis at Q. Graph of y = ax + b O y x y = ax + b Q The graph cuts the y-axis at Q. x-intercept a b -

Graph of y = ax + b x-intercept y-intercept The graph of a linear function is a straight line. It has only one x-intercept and one y-intercept.

Graph of y = ax + b a 0 a 0 O y x O y x The value of y increases as x increases. The value of y decreases as x increases.

Follow-up question The figure shows the graph of the linear function y = f(x). Write down the x-intercept and the y-intercept of the graph. x y 2 1 1 2 3 4 4 2 2 4 ∵ The graph cuts the x-axis and the y-axis at (2, 0) and (0, 3) respectively. y = f(x) (0, 3) (2, 0) ∴ x-intercept = 2 and y-intercept = 3