Take your time to play around with the variables, and get a better feel for how changing each of the variables effects the nature of the function. To get a better look at exponential functions, and to become familiar with the above general equation, visit this excellent graphing calculator website here. There is no value for x we can use to make y=2.Īnd that's all of the variables! Again, several of these are more complicated than others, so it will take time to get used to working with them all and becoming comfortable finding them. For our other function y = 2 x 2 y=2^x 2 y = 2 x 2, k=2, and therefore the horizontal asymptote equals 2. This makes sense, because no matter what value we put in for x, we will never get y to equal 0. Take for example the function y = 2 x y=2^x y = 2 x: for this exponential function, k=0, and therefore the "horizontal asymptote" equals 0. "k" is a particularly important variable, as it is also equal to what we call the horizontal asymptote! An asymptote is a value for either x or y that a function approaches, but never actually equals. If "k" were negative in this example, the exponential function would have been translated down two units. Let's compare the graph of y = 2 x y=2^x y = 2 x to another exponential equation where we modify "a", giving us y = ( − 4 ) 2 x y=(-4)2^x y = ( − 4 ) 2 xīy making this transformation, we have translated the original graph of y = 2 x y=2^x y = 2 x up two units. ![]() ![]() But, so you have access to all of the information you need about exponential functions and how to graph exponential functions, let's outline what changing each of these variables does to the graph of an exponential equation. In this lesson, we'll only be going over very basic exponential functions, so you don't need to worry about some of the above variables. The above formula is a little more complicated than previous functions you've likely worked with, so let's define all of the variables.Ī – the vertical stretch or compression factorĭ – the horizontal stretch or compression factor Y = a b d ( x − c ) k y=ab^ k y = a b d ( x − c ) k Now that we have an idea of what exponential equations look like in a graph, let's give the general formula for exponential functions: That is because as x increases, the value of y increases to a bigger and bigger value each time, or what we call "exponentially". In the case of y = 2 x y=2^x y = 2 x and y = 3 x y=3^x y = 3 x (not pictured), on the other hand, we see an increasingly steepening curve for our graph. That is why the above graph of y = 1 x y=1^x y = 1 x is just a straight line. As you can see, for exponential functions with a "base value" of 1, the value of y stays constant at 1, because 1 to the power of anything is just 1. Second, weĪpply our results to the analysis of ERG models.The table of values of y = 1^x and y = 2^xĪbove you can see three tables for three different "base values" – 1, 2 and 3 – all of which are to the power of x. ![]() We discuss its relevance to maximum likelihoodĮstimation, both from a theoretical and computational standpoint. The statistical and geometric properties of the corresponding extendedĮxponential families. $k$-dimensional exponential families of distribution with discrete base measureĪnd polyhedral convex support $\mathrm$ is a geometric object that plays a fundamental role in deriving ![]() Relate to the broader structure of discrete exponential families. The issues associated with these difficulties Graph (ERG) models, especially in connection with difficulties in computing Network data, and considerable interest in the class of exponential random Fienberg and 2 other authors Download PDF Abstract: There has been an explosion of interest in statistical models for analyzing Download a PDF of the paper titled On the Geometry of Discrete Exponential Families with Application to Exponential Random Graph Models, by Stephen E.
0 Comments
Leave a Reply. |