A function is a rule that maps a single number to another single number Functions can be written in different ways y = 2x + 3 f(x) 2x + 3 f : x 2x + 3 Using the function f (x) x2 , one number input gives one number output However, the function f (x) one input number gives 2 outputs and is therefore not a function When plotting on axes the vertical axis is labelled f (x) or f : x not y

Determine which of these mappings are functions: Qwertyuiiop[]#’;’ Determine which of these mappings are functions: f(x) ’2x - 3 FUNCTION NOT A FUNCTION FUNCTION NOT A FUNCTION f(x)’ 1 x - 1 FUNCTION NOT A FUNCTION f(x) ’ FUNCTION NOT A FUNCTION f(x)’ x2 + 3 FUNCTION NOT A FUNCTION f(x) ’ the greatest integer less than or equal to x FUNCTION NOT A FUNCTION f(x)’ the height of a triangle whose height is x Next Activity

Correct Return

WRONG!!! Return

Translating Functions Principles If y = x is the function f(x) x Then f(x) + 3 is the function f(x) x + 3 1 3 4 5 2 7 6 - 7 -1 -3 -4 -5 -2 -6 -7 1 3 4 5 2 7 6 - 7 -1 -3 -4 -5 -2 -6 -7 y = x + 3 f( x) + 3 y = x f( x) We have seen from the equations of straight lines that from the general equation y = mx + c, the integer c has an effect on where a straight line crosses the y axis So the adding or subtracting of an integer translates the graph up or down f(x) 1 3 4 5 2 7 6 - 7 -1 -3 -4 -5 -2 -6 -7 1 3 4 5 2 7 6 - 7 -1 -3 -4 -5 -2 -6 -7 f(x) + 3 This is the same for any function This can also be described as a translation

Translating Functions Principles There are other translations that can be performed on graphs: y = x2 y = (x-2)2 The equation (x – 2)2 = 0 has root x = 2 Compare the equation y = (x – 2)2 with the graph y = x2 2 This can also be described as a translation So to summarise these two translations: The graph of y = f (x) + a is the graph of y = f(x) translated by The graph of y = f (x - a) is the graph of y = f(x) translated by

Stretching Functions We will use the sine wave as an example y = sinx Now see what happens when we double the function y =2 x sinx or more precisely written y =2sinx The graph is stretched to twice its size in the x direction Now see what happens when we double the x value before finding the sine y =sin (2 x x) The graph is squashed to half its size in the y direction or more precisely written y =sin2x

Summary Graph Description Translation   Graph Description Translation Stretch by a scale factor a parallel to the y axis   Stretched by scale factor parallel to the x axis Reflection in the x axis of Reflection in the y axis of