Is the rules I got, and the logic to how I got them mathematically solid? Consider the function y(x) = -2(-4x – 2) 2 + 1įirst I applied the reflections, the horizontal shrink by a factor of 1/4, and the vertical stretch by a factor of 2. Vertical translations apply to the parent function as a whole, so they go last.Ĭ. Horizontal translations apply operations to the x-values, so they go first. I discern the order of the translations the same way. In summary, this is the order I’ve come up with at this point for applying transformations. They both involve multiplying the same parts of the function, and by the commutative property of multiplication, it doesn’t matter the order in which I do the multiplication. Then, I came to the conclusion that I could apply not just reflections but vertical/horizontal stretching at the start as well. A reflection across the y-axis applies a negative value to the x values only. Reason being that because I shifted to the right one and then reflected across the y-axis, I was actually reflecting not the x values, but the x – 1 values. So my rule of applying transformations like the order of operations falls apart.įrom here it occurred to me that I should apply reflections first. To get the right answer, I would have to apply the reflection first and then shift 1 unit to the right even though the part that causes the shift is inside the parenthesis. If I do it that way, I get the wrong answer. Since that negative is inside the radical, it results in a reflection about the y-axis. Then, I would multiply that result by a negative. If I were to plug in an x, I would first subtract a 1, that corresponds to a shift 1 unit to the right. Now, let’s do the same thing to g(x) = √(-(x – 1)). Apply that to the whole graph, and I have my transformed function.ī. After squaring, I would add 3 which corresponds to a shift 3 units up. First, I would add 1, which corresponds to a horizontal shift one unit to the left because it’s inside the parenthesis. When it comes to how I would apply the transformations, I think about how I would do the operation if I were to plug in an x in the transformed function, via the order of operations. The parent function is p(x) = x 2, so I start with a graph of that. I have to then manipulate the parent function to get the graph of the new function. I am given the graph on an xy-plane, and I am given the new function. I’ve been thinking about the order in which to apply the transformations to a graph when transforming its parent graph. Specifically. What order do we apply function transformations?īy transformations, I mean stuff like horizontal/vertical stretching/shrinking and translations. The question came from Mario in early September, working through how to determine the appropriate transformations to graph a given function: Here, the focus will be on examples and alternate approaches next week, the underlying reasons. A recent discussion brought out some approaches that nicely supplement what we have said before. Transformations of functions, which we covered in January 2019 with a series of posts, is a frequent topic, which can be explained in a number of different ways.
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