When we multiply a function by a positive constant, we get a function whose graph is stretched or compressed vertically in relation to the graph of the original function. Each change has a specific effect that can be seen graphically. We can transform the inside (input values) of a function or we can transform the outside (output values) of a function. We now explore the effects of multiplying the inputs or outputs by some quantity. Is the function f ( s ) = s 4 + 3 s 2 + 7 f ( s ) = s 4 + 3 s 2 + 7 even, odd, or neither? Graphing Functions Using Stretches and CompressionsĪdding a constant to the inputs or outputs of a function changed the position of a graph with respect to the axes, but it did not affect the shape of a graph. For a function g ( x ) = f ( x ) + k, g ( x ) = f ( x ) + k, the function f ( x ) f ( x ) is shifted vertically k k units. In other words, we add the same constant to the output value of the function regardless of the input. The simplest shift is a vertical shift, moving the graph up or down, because this transformation involves adding a positive or negative constant to the function. One simple kind of transformation involves shifting the entire graph of a function up, down, right, or left. There are systematic ways to alter functions to construct appropriate models for the problems we are trying to solve. One method we can employ is to adapt the basic graphs of the toolkit functions to build new models for a given scenario. Often when given a problem, we try to model the scenario using mathematics in the form of words, tables, graphs, and equations. Graphing Functions Using Vertical and Horizontal Shifts In this section, we will take a look at several kinds of transformations. In a similar way, we can distort or transform mathematical functions to better adapt them to describing objects or processes in the real world. But what happens when we bend a flexible mirror? Like a carnival funhouse mirror, it presents us with a distorted image of ourselves, stretched or compressed horizontally or vertically. When we tilt the mirror, the images we see may shift horizontally or vertically. To negative 3 times g of x.We all know that a flat mirror enables us to see an accurate image of ourselves and whatever is behind us. Red graph right over here is 3 times this graph. 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. 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.
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