Combining functions shifting and scaling graphs
A function f x is called even if its value at x is the same as its value at -x. An even function has a graph that is symmetric about the y- axis: A function f x is called odd if its value at -x is the negative of its value at x. An odd function has a graph that is symmetric about the origin: Equivalently, you can reflect this part of the graph about the origin, point by point, as is shown in the sketch combining functions shifting and scaling graphs.
Shifting and Scaling The three main transformations we will consider, horizontal shifts, vertical scalings and vertical shifts, are described here with examples. Understanding these examples will help in the exercises to follow. The graph of f x - c is the graph of f x shifted horizontally c units. The graph of kf x is the combining functions shifting and scaling graphs of f x scaled vertically. If k x -axis and becomes less steep. In addition, if k x axis. Effect of absolute value When a function is contained within absolute combining functions shifting and scaling graphs symbols, first consider the graph of the same function without the absolute value, and then flip the parts of the graph that are below the x axis.
If f x y value positive so those points below the x axis are flipped above the x axis. Examples of Graphing Techniques. Click on "New" to generate a problem with this type of effect. Click on "Graph it" to see the graph of the new function. Choose the type of function you want to graph by clicking on one of the four "Function type" buttons. To move the graph horizontally or vertically, click on the "Horizontal" or the "Vertical" button, then use the mouse to drag the graph to the right place.
To flip the graph, click on "Flip" To scale the graph, click on one of the four Scaling factors to apply. This is cumulative so scaling by 2 and then by 3 scales the original function by 6. When you're done, click on "Check it". To get a new function click combining functions shifting and scaling graphs on one of the four "Function type" buttons. If you want to give up, you can click on "Graph it" to see what the graph looks like.
Here we will try to make clear how to guess the graphs of functions that were obtained by applying transformations to known functions. We will not cover functions that are obtained by combining more functions together, since guessing such graphs is difficult, often impossible. The situation we cover here is as follows: We have one elementary function and we apply one or more transforms to it. If you already looked at the note on order of evaluation and the algebraic visualization of functions, it will make sense to you when we say that here we will not guess graphs of functions that are not one chain but have more "roots".
Incidentally the order of operations may come handy here as well. An important part of this section will be the list of transformations and how they affect the graph. Since here we are interested in practical application, we will make a "wrong" list in the sense that one transformation will be covered several times, split into special cases. For instance, the scaling c x will be split into four cases, since when working problems, it is intuitively easier.
More details on transformations can be found in the section Operationsbelow we just symbolically express their effect. At the end of this section we will also show how to guess the effect of some functions that do not belong among the traditional transformations, but an experienced student might find the note helpful. Every transformation works in the same way, by replacing. Every transformation also comes in two flavours, they can be applied to the argument or to the value that is, to a function.
We will represent this replacement procedure by an arrow. Guessing algorithm Step 1. Identify the basic elementary function whose graph is the basis for the function you face. Sketch the graph of the basic function, then modify it according to all transforms that are applied to the argument last to first with respect to the order of operations, see below.
Take the graph obtained in Step 2 and modify it according to all transforms that are applied to the value first to last with respect to the order of operations, see below. One of the most difficult parts for a student is to recognize the correct order in which the transformations should be applied. Typicaly, there would be two transformations applied to the argument, the question then is which one comes first. Fortunately, there is an easy way to find out. If you start with x and apply the replacements as showed above by arrows in the same order in which you modify the graph, you should get the argument in the given function.
If you get something else, your order was wrong. This rule also applies if more transforms are applied. We will now try the replacement business for both: There is a longer but perhaps a bit more revealing way of showing how the transformations that is, replacements affect the starting function. We will show it for the first correct variant.
If we think of the order in which we calculate the given expression, the transformations should be applied "last to first". Last to first is a rule you can remember, but only for the argument.
Tranformations that apply to values are done in exactly the opposite order, first to last. First we identify the core function. Indeed, all other operations appearing in f are transformations. This means shrinking the graph three times horizontally towards the origin and shifting it to the right by 2. If we shrink first, shift next, the replacement chain will go.
Thus we do this:. There is sort of a check whether we got it right. Note that if we did the two transformations in the wrong order, we would get something different:. Now it is time to apply transformations to the value. There are three of them, vertical scaling by 4, shift down by 1 and mirroring using absolute value. The absolute value is calculated last, so it will be also the last one. The "first to last rule" says that we should scale the graph by 4 vertically, then shift down by 1 and then mirror.
We check that the replacements work:. It helps to keep track of some important points during those steps, we may try to substitute zero or check on zero points. Here we will cover three more transformations.
They are rather advanced and in most calculus courses they do not work with them. We put them here because they are sometimes useful to know they helped me a few times and the section would not feel complete without them. Here is the situation: We know the graph of some function f. The square power combines two effects. One, it changes all negative values into positive, so we should start by mirroring the negative parts of the graph of f up above the x -axis.
The second effect is that of scaling, but unlike the above cases, the scaling is not by a common factor. Briefly, the larger the value of fthe more it gets enlarged by squaring. This suggests the steps:. Then do the following. On the left we show the graph of some funkce fwe also mark the effect of mirroring and the level 1.
The effect of other integer powers greater than 1 is similar. Even powers work exactly the same, just the stretching and squeezing effect is greater for powers greater than 2. How does the graph of the root of f look like? The square root can be only applied to functions that do not have negative values. Here we have to remember that square root makes numbers larger than 1 smaller, but enlarges numbers smaller than 1.
On the left we show the graph of some function fwe also mark the level 1. The effect of other roots is similar. If n is an even positive integer, then the n -th root works just like the square root. If n is odd, then the n -th root has similar effect, but now it can be applied to all functions and their negative parts get affected in exactly the same way as we transformed the positive parts.
The reciprocal can be only applied to functions that are not zero. We know that large numbers are turned into small ones and vice versa by the reciprocal value, but they keep their signs, which suggests what we should do.
Divide the graph into horizontal strips. The parts of the graph that are in the first two strips exchange places.
Points that were further will be further in the following way: The pieces of the graph that went to infinity will now approach 0. The parts of the graph that approached 0 will be sent to infinity. The two strips in the lower half will be treated exactly as in 3. Putting the ideas of 1 - 3 together we now can guess also negative integer powers and roots.
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