![]() ![]() values were related to the corresponding -values of the original function. in thick style and in thin style in a Zoom Standard window. abs (x) and Y abs (x) + 3 in the Y editor. ![]() sends all points of a graph the same distance in the same direction. Red graph right over here is 3 times this graph. Lesson 3.2: Translations and Combined Transformations. It looks like weĪctually have to triple this value for any point. X looks like it's about negative 3 and 1/2. When we flip it that way, this is the negative g of x. Here we would call- so if this is g of x, Its mirror image, it looks something like this. Image but it looks like it's been flattened out. Would have actually shifted f to the left. Little bit counter-intuitive unless you go through thisĮxercise right over here. Is shifting the function to the right, which is a When I get f of x minus 2 here-Īnd remember the function is being evaluated, this is the X is equal to f of- well it's going to be 2 less than x. g of whatever is equal to theįunction evaluated at 2 less than whatever is here. See- g of 0 is equivalent to f of negative 2. Is right there- let me do it in a color you can This point right over there is the value of f of negative 3. So let's think aboutĪrbitrary point here. Similar to the other one, g of x is going to X is, g of x- no matter what x we pick- g of x And we see g of negativeĤ is 2 less than that. Is f of x in red again, and here is g of x. But if you look atĮqual to f of x plus 1. Try to find the closest distance between the two. Of an optical illusion- it looks like they So it looks like if we pickĪny point over here- even though there's a little bit Write this down- g of 2 is equal to f of 2 plus 1. And we see that, at leastĪt that point, g of x is exactly 1 higher than that. Hope I didn't over explain, just proud of what I made tbh so 5*f(x) would make a point (2,3) into (2,15) and (5,7) would become (5,35)ī will shrink the graph by a factor of 1/b horizontally, so for f(5x) a point (5,7) would become (1,3) and (10,11) would become (2,11)Ĭ translates left if positive and right if negative so f(x-3) would make (4,6) into (7,6) and (6,9) into (9,9)ĭ translates up if positive and down if negative, so f(x)-8 would make the points (5,5) and (7,7) into (5,-3) and (7,-1)Īlso should note -a flips the graph around the x axis and -b flips the graph around the y axis. So for example if f(x) is x^2 then the parts would be a(b(x+c))^2+dĪ will stretch the graph by a factor of a vertically. Then if m is negative you can look at it as being flipped over the x axis OR the y axis.įor all other functions, so powers, roots, logs, trig functions and everything else, here is what is hopefully an easy guide. ![]() We can work around this by factoring inside the function.Yep, for linear functions of the form mx+b m will stretch or shrink the function (Or rotate depending on how you look at it) and b translates. This format ends up being very difficult to work with, because it is usually much easier to horizontally stretch a graph before shifting. To solve for x, we would first subtract 3, resulting in a horizontal shift, and then divide by 2, causing a horizontal compression. ![]() What input to g would produce that output? In other words, what value of x will allow g\left(x\right)=f\left(2x+3\right)=12? We would need 2x+3=7. When we write g\left(x\right)=f\left(2x+3\right), for example, we have to think about how the inputs to the function g relate to the inputs to the function f. Horizontal transformations are a little trickier to think about. In other words, multiplication before addition. Given the output value of f\left(x\right), we first multiply by 2, causing the vertical stretch, and then add 3, causing the vertical shift. When we see an expression such as 2f\left(x\right)+3, which transformation should we start with? The answer here follows nicely from the order of operations. For example, vertically shifting by 3 and then vertically stretching by 2 does not create the same graph as vertically stretching by 2 and then vertically shifting by 3, because when we shift first, both the original function and the shift get stretched, while only the original function gets stretched when we stretch first. When combining transformations, it is very important to consider the order of the transformations. Take note of any surprising behavior for these functions. Given the toolkit function f\left(x\right)=, graph g\left(x\right)=-f\left(x\right) and h\left(x\right)=f\left(-x\right). ![]()
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