M1410105Part1

M1410105Part1 - (Section 1.5: Analyzing Graphs of...

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(Section 1.5: Analyzing Graphs of Functions) 1.38 SECTION 1.5: ANALYZING GRAPHS OF FUNCTIONS PART A: GRAPHING y = f x ( ) The graph of f , or the graph of y = f x ( ) , in the standard xy -plane consists of all points [representing ordered pairs] of the form x , f x ( ) ( ) , where x is in the domain of f . In a sense: Graph of f = x , f x ( ) ( ) x Dom f ( ) { } . Here, as we typically assume throughout Precalculus, … x is the independent variable, because it is the input variable. y is the dependent variable, because it is the output variable. Its value (the function value ) typically “depends” on the value of the input x . Technical Note: Even for a constant function f where, say, f x ( ) = 3 , we refer to y as the dependent variable, even though (informally speaking) the value of y is always 3 and does not really “depend” on the value of x in the traditional sense. “3” does technically represent a function of x . Example Consider the graph of f x ( ) or y = x . See p.68 . Note: y and f x ( ) are interchangeable here. The domain of f is 0, ) , the nonnegative reals. The input x = 9 corresponds to the point 9, f 9 ( ) ( ) , or 9,3 ( ) , on the graph. The input x = 9 is not “legal” in that it does not lie in the domain of f . There is no corresponding point on the graph.
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(Section 1.5: Analyzing Graphs of Functions) 1.39 Here is the graph of f (with a few points indicated): Warning: Clearly indicate any endpoints on a graph, such as the origin here. The lack of a right endpoint on our graph implies that the graph extends beyond the edge of our figure. We want to draw graphs in such a way that these extensions are “as one might expect.” Technical Note: The x between the 4 and the 9 on the x -axis represents a generic x -coordinate in the domain. Some may prefer to use x 0 (called x sub zero” or the more British “ x naught”) to represent a particular or fixed x -coordinate. In these notes , we may sometimes be a bit sloppy and use x when perhaps x 0 would be more appropriate. The Point-Plotting Method for Graphing a Function This is when you choose a bunch of x values in the domain, find their corresponding f x ( ) values, plot the corresponding points x , f x ( ) ( ) , and connect the dots “nicely.” We will generally avoid this method by appealing to properties of the function, but it is available as a last resort!
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(Section 1.5: Analyzing Graphs of Functions) 1.40 PART B: THE VERTICAL LINE TEST (VLT) An equation in x and y describes y as a function of x , and we can then say y = f x ( ) Its graph passes the VLT in the standard xy -plane, meaning that there is no vertical line that intersects the graph more than once (i.e., there is no input x that yields more than one output y ). Example
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This note was uploaded on 09/08/2011 for the course MATH 141 taught by Professor Staff during the Fall '11 term at Mesa CC.

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M1410105Part1 - (Section 1.5: Analyzing Graphs of...

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