basic_graphs

basic_graphs - Basic graphs Graphs of functions of the form...

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Graphs of functions of the form = ( ) f x x r . First we consider the graph of the equation = y x 2 . If = ( ) f x x 2 , then = ( ) f - x ( ) f x for all real numbers x , that is, changing the sign of an input number x for the function f does not change the value of f. For example, = f - 3 2 f 3 2 = 9 4 . This leads to the fact that the graph of = y x 2 is symmetrical about the y axis . In general, a function that satisfies the condition that = ( ) f - x ( ) f x for all numbers x in the domain of f is called an even function . The graph of an even function is always symmetrical about the y axis. The graph of = y x 2 is an example of a parabola . In general, a parabola is a curve consisting of all points that lie at an equal distance from a specified point ( called the focus of the parabola ) and a specified line ( called the directrix of the parabola ). The focus of the parabola given by = y x 2 is the point , 0 1 4 and the directrix is the horizontal line with equation = y - 1 4 . As an example, we can check that the point ( ) , 1 1 is equidistant from , 0 1 4 and = y - 1 4 . The point ( ) , 1 1 clearly lies at a distance of 5 4 vertically above the line = y - 1 4 . The difference in the x coordinates of the two points , 0 1 4 and ( ) , 1 1 is 1 unit while the difference between the y coordinates is 3 4 . The distance between the two points can then be calculated using Pythagoras' theorem and is = + 1 2 3 4 2 + 1 9 16 = 25 16 = 5 4 . Now consider the graph of the equation = y x 3 . If = ( ) f x x 3 , then = ( ) f - x - ( ) f x for all real numbers x , that is, changing the sign of an input number x for the function f just changes the sign of ( ) f x . For example, = f 5 4 125 64 = 1.953125 and = f - 5 4 - 125 64 = is symmetrical about the origin .
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basic_graphs - Basic graphs Graphs of functions of the form...

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