![]() Let’s look at actual graphs of a specific function. To put it another way, the shrink in the second version also moved the starting point of the graph I drew (by shrinking the empty space), so I had to shift it less to get to the destination graph. The other shrinks first, and then shifts-but not as far, since the shrink reduced the distance it has to go: The first takes the graph of f and moves it left then shrinks it: First, let's visualize what each pair of transformations does. There are a couple things to notice here. As we’ll see later, some books teach this form as their routine method I approve of that because this order of transformations works better for many students. We would do this because of what the first example showed, that this order resulted in the factored form, with parentheses, so we used that form. Shift b/a units to the left: f(a(x+b/a)) = f(ax+b) ![]() Shrink horizontally by a factor of a: f(ax) We can write f(ax+b) as f(a(x+b/a)), factoring out the a, and then do this: But in fact we COULD do the two transformations in the other order, if we change the particular amounts. When I want to be sure of the order, I always take it step by step like this. Shrink horizontally by a factor of a: f(ax+b) If instead we first do the shift, changing f(x) to f(x+b), and THEN do the shrink, we replace x in x+b with ax, and get f(ax+b), which is what we want. Here, shrinking first, changing \(f(x)\) to \(g(x) = f(ax)\), and then shifting, \(h(x) = g(x + b) = f(a(x+b))\), didn’t result in the desired function \(h(x) = f(ax+b)\). ![]() As I said here, transformations can be applied in any order, but changing the order changes the result, so the trick is to find the order that results in the desired transformed function. ![]() I’ll also be emphasizing later some details on what each transformation does to the graph. I will be reiterating the key idea several times: the horizontal transformations (which affect the input to the function) should be thought of as replacing x with a new expression. So this order of doing those particular transformations is wrong. That is NOT what we are looking for it's equal to f(ax+ab). If we then apply a horizontal shift (translation) b units to the left, we would be REPLACING x in f(ax) with x+b, and we'd get f(a(x+b)). Suppose we first do the horizontal shrink f(x) -> f(ax). What I do is to explicitly write the steps, one at a time. The order makes a difference in how you get there. But in this case, you are asking in which order to do them in order to transform f(x) into a specific goal, f(ax+b). You can perform transformations in any order you want, in general. Since this is a favorite topic of mine, I answered: Good question! I haven't seen this treated well in textbooks, either, but it's an important topic. Which comes first? (And the example he gives is the hardest case.) Ozgur has learned about each individual transformation (respectively, vertical and horizontal reflections, vertical and horizontal stretches or shrinks, and vertical and horizontal shifts) but now wants to be able to read a function and determine the correct sequence of transformations. I've looked in many textbooks and have been unable to find an answer. We’ll spend most of our time with the following question, from 2004: Order of Transformations of a FunctionĬould you please tell me in what order I would perform transformations such as -f(x), f(-x), af(x), f(ax), f(x)+a, f(x+a) if two or more were to be applied to f(x)? As an example, if I had f(ax+b) would I do the As we do this, we will develop a deeper understanding of how each transformation works, and how they interact. Now we can look at cases where two or more transformations are combined. All our examples involved only a single transformation. Last time we looked at questions about how to shift, stretch, or flip a graph by changing the equation of a function.
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