Transcript for:
Parent Functions and Transformations

in this video we're going to be talking about parent functions and transformations let's jump right in all right so let's start things off by making some basic observations about these four parent functions in general parent functions are the simplest form of a given family of functions the first family functions we're going to look at are constant functions the quote unquote parent function of all constant functions is f of x is equal to 1. and just to interpret what this means regardless of what x value you input the output will always be one and we can see this on the graph regardless of what x value you choose the output which is y is always one and because of this constant functions will always be horizontal lines since the y value never changes it never goes up or down the next family of functions we're going to look at are linear functions like the name suggests these are also lines but they're not necessarily horizontal like constant functions the parent function in this case is f x is equal to x and just to interpret this if you input a value of x then the output will be that same value so for example let's say we set our input x to be equal to 1 then our output will be 1 as well the most important characteristic of linear functions to know is that every unit increment in x will result in the same change in y and to put this characteristic into more mathematical or perhaps more familiar terms we refer to this as having a constant slope all right now let's move on to the absolute value functions in this case the parent function is f of x is equal to the absolute value of x and from the graph we can see that the range of this function that is the output is always positive because absolute value by definition is just the magnitude of a number you discard its sign and so as you can see on this graph positive x values correspond to positive y values but then negative x values correspond to positive y values as well because we're discarding the sign and because of this all absolute value functions will be v shaped the fourth and final family functions we're going to be talking about our quadratic functions in this case our parent function is f of x is equal to x squared so you input a value of x and the output will be that value squared for this particular function y is always positive because the square of a number whether it's positive or negative will always be positive for quadratic functions every unit increment in the positive x direction will result in an increasing change in y and as a result all quadratic functions will be u-shaped now these are four relatively simple types of functions and we'll be getting into more of them later on in this course now let's move on to transformations which will be the focus of the rest of this video essentially transformations are ways we can alter or manipulate these basic parent functions using transformations we can change the position of our function on the x y plane we can shift in the horizontal direction or the vertical direction and these types of shifts are known as translations now we could also change the orientation of these functions in the case of the absolute value or quadratic functions we can alter whether they open up or down and things like that and changes in orientation are known as reflections and we can also alter the size and shape of a function by stretching or shrinking it for now we'll focus on translations and reflections and i'll cover stretching and shrinking in a future video let's start off by going over translations like we mentioned in the previous slide translations are a type of transformation that involve moving or shifting a function horizontally or vertically in the x y plane so we can break down translations into vertical shifts or vertical translations and horizontal shifts or horizontal translations vertical shifts are a little bit easier so let's start there essentially if you have some function f of x and you define g of x to be f of x plus k then what g of x is is f x shifted k units up and similarly if you define g of x to be f of x minus k then g of x is f of x shifted k units down so when we're adding the constant we're shifting up and when we're subtracting a constant we're shifting down so let's do an example of this let's say f of x is equal to x squared so we're dealing with the parent function for a family of quadratic functions okay so here's f of x graphed and now let's consider what g of x which we can define as f of x plus two so x squared plus two let's consider what g of x would look like in relation to f of x remember since we're translating the shape of the function is not going to change but the position is we're clearly dealing with some sort of vertical shift here clearly we're dealing with some sort of vertical shift here g of x is in the form f of x plus k and k is 2 and so g of x is going to be f x shifted 2 units up is going to look something like this and if you want to think of it this way for every point x comma y on our curve f of x there is a corresponding point on the curve g of x where the x value is the same but the y value is the y value for f of x so y plus 2. and so this corresponds to the shifting of the graph two units upward now visually the two in this sort of diagram is the vertical distance between the two curves at any x value now let's say we have a third function h of x which is defined as f of x minus three so x squared minus three this is still a vertical shift but now we're shifting in the negative y direction since k is negative specifically we're shifting f x three units down and so h of x would look just like f of x except now all its points would be shifted three units down so if we think about starting at the vertex the vertex is zero zero and f of x and so h of x would have a vertex at zero negative three so it would look something like this and that's really all there is to know about vertical shifts now it's going to do horizontal shifts so once again if we have a function f of x and we define g of x to be f of x minus h then all g of x is is f x shifted h units to the right and if we have g of x defined as f of x plus h then g of x is just f of x shifted h units to the left now the one thing that's a little bit not intuitive about horizontal shifts is that the direction of the shift is opposite to the sign so for example if we consider this second case where we have f of x plus h we typically associate the plus sign with moving to the right but it turns out that this actually means f of x is shifted h units to the left so that's just something to keep in mind let's start off by considering the case where we have the function g of x and we define it as being f of x minus two so in other words if we plug in x minus two into our function f we get x minus two whole squared now just based on our definition because we can see g is in the form f of x minus h where h is two we know that the function is going to shift two units to the right but let's just think a little bit about the intuition behind that let's consider when f of x is equal to one now f of x is equal to x squared and so f of x is going to be equal to one when x is plus or minus one and so the two points we're looking at are negative one comma one which is here and one comma one which is here now let's consider the points where g of x is equal to one remember g of x is x minus two squared and we can see that x minus two whole squared is equal to 1 when x is equal to 1 in the case you get 1 minus 2 which is negative 1 squared equals 1 or when x is equal to 3 3 minus 2 is 1 and 1 squared is 1. and so the points on the curve g of x where g of x is equal to one are one one and three one and so if i plot those two points the point one one was the point we plotted before so i'll just indicate it with a smaller blue circle inside that white circle and the point three one is this point right here and notice something special about these two points let's first consider these two one comma one is just the point negative one one shifted to the right two units and similarly three comma 1 is the point 1 comma 1 shifted to the right 2 units and if you were to plug in some more points you would see that g of x looks roughly like this and my x y plane isn't perfectly to scale and that and that's why the shapes of f x and g of x look slightly different but in reality the shapes of these two functions are exactly the same since all we're doing is changing the position specifically shifting f x two units to the right will give us g of x and so in this case if we're looking visually all the two represents is the horizontal distance from the pink curve to the corresponding point on the blue curve at any given y value and of course we can do a shift in the opposite direction as well let's say we have a function h of x which we define to be f of x plus three so h of x is x plus three whole squared and since in this case h is positive we're going to be shifting to the left and specifically since h is 3 we're going to be shifting f 3 units to the left to get h and so h of x is going to look roughly like this once again if you just want to think about the vertex for f of x the vertex is at zero zero and for h of x it's been shifted three units to the left so it's now at negative three comma zero and that's really all there is to horizontal shifts now let's move on to reflections so while translations involved changing the position of the graph of a function reflections involve changing that graph's orientation so this could involve flipping the graph of that function over a line of reflection so that line could be the x-axis or the y-axis or maybe the line y equals x and things like that but for now we're just going to focus on reflecting over the x-axis and reflecting over the y-axis let's start off with reflecting over the x-axis if you have some function f of x and you define and you define g of x to be negative f of x all g of x's is f of x flipped over the x axis so let's stick with f of x from last time we have f of x is equal to x squared so here we have f of x graphed as you can see it's a parabola that's opening upward and now let's say we create a function g of x which is defined as negative f of x so negative x squared now since g of x is in this form negative f of x we know it's just going to be f of x flipped over the x-axis which is this line right here so just imagine you take f of x and you flip it over that line and so g of x is going to look something like this once again we're preserving the shape but we're changing the orientation the parabola g of x opens downward while the parabola f x opens upward in terms of the intuition behind this consider any point on the curve f x right let's say we look at this point again which is the point one one now let's consider the corresponding point on the curve g of x so we're going to plug in the same input so we're looking for g of one and g of one is just negative of f of one f of one is one and since we have that negative sign g of one will be negative one so it's this point right here and so ultimately what we see is that all the points on the pink curve get flipped over the x-axis to form the blue curve that's really all there is to reflecting functions over the x-axis now let's talk about reflecting over the y-axis so if you have some function f of x and you define g of x to be f of negative x then all g of x really is is f x flipped over the y axis all right this time let's say f of x is equal to x cubed all right so this is what f x looks like it's a cubic function now imagine that we have a function g of x which is defined as f of negative x so whenever we see x we're going to replace it with negative x so in this case we would get negative x cubed so this function is in the form of g of x is equal to f of negative x and whenever a function is in this form we know it's f of x flipped over the y axis so it's going to look something like this so in order to understand the intuition behind this let's consider the point 1 comma 1 which is on the curve f x and now let's think about the corresponding point on the curve g of x the corresponding point on g of x is going to be negative 1 comma 1 because when we pass in negative 1 to g of x we get g of negative one which is equal to f of negative of negative one which is equal to f of one so we see that f of one is equal to g of negative one but also in general we can see that f and g have the same y values when their x values are of opposite signs and this explains why g is f reflected over the y axis and so hopefully this intuition made sense alright that about wraps it up for this video if it did help you at all please be sure to leave a like and if you want to be notified when i post the rest of the videos in this course make sure to subscribe thanks for watching