{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

basic_graphs

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

This preview shows pages 1–2. Sign up to view the full content.

Basic graphs 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 = - 1.953125.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}