Horizontal & Vertical Translations of Functions • [1.1] PRE-CALCULUS 12

hey there in this video we are going to look at how replacing x with X plus or minus a number or Y with Y plus or minus a number in the equation of a function

causes a horizontal or vertical translation in the graph of that function before we start talking specifically about translations we should recognize that in general the

word transformation when you're talking about functions refers to something that changes the equation and a corresponding change in the graph where the location

or shape or orientation or any combination of those are changed in mathematics when you talk about functions the term translation is a kind

of transformation that refers to a shift in the graph of a function where it moves to a new location but the shape and orientation don't change a

translation can be vertical like that or it can be horizontal like that or you could have some combination of the two of those things and it can end

up anywhere first we're going to look at vertical translations and as our base function we're going to use yals absolute value of x the function you saw a minute ago in the graph that I was

moving around and we're going to make two changes here we're going to take that base function and change it to Y = absolute value of x - 3 and to Y =

absolute value of x + 4 and see what happens with the values in the table here now before we move forward it's worth noting that you could actually write this in a different format you

could write this as y + 3 equals absolute value of x and you can write this one as y - 4 that + 4 could go over

there as minus 4 absolute value of x now that's along the lines of what I alluded to at the beginning of the video when I said that we were going to be replacing

y with Y plus or minus a number like that but we are going to stick with the given format because it's easier for calculating those values in the table

here the yv values now if you go ahead and substitute the X values we're using the same X values in each of those tables and you substitute it in for X in

each of those here's the yv values that you get now now in the first table it's the values that you expect just the absolute value of x right if x is

positive it stays positive and if x is negative it becomes positive the other ones what's happening is that same thing is happening say for example here you

substitute in two into this you put two in here absolute value of two is two but then you're going to subtract three after you've worked out that absolute value which leaves you with minus1 the

same goes here say for this minus 3 here you put a a -3 in there absolute value is positive3 and then you subtract three so you get zero in each of those cases

you're subtracting three after you work out the absolute value or in other words after you work out these same values here that's what that's going to produce and then you're just going to subtract

three from all of them so what you'll notice here is these values are the same as these values except they're three lower all the Y values have been reduced

by three and you see the same thing in the second table there that all of these are for example again if you put a three in Absol of three is three and plus 4 to

that gives you a seven let's do one on the other side here - 1 you put negative 1 in there absolute value of negative 1

is positive 1 + 4 is five and in a similar way these y values are these y values but four greater you've added

four after you've worked out those absolute values so the change that you see there in those tables for the middle one for this one you have that the Y

values are three lower for the same X values and in the

second one there the Y values are for higher we're going to graph the those three functions now you can go ahead and

graph the points in the table and see what the three graphs look like when we go on do that in the interest of time here this is the result that you're going to get you're going to get that

vshape three different times there but it's going to be in a different location it's the same shape but in a different location there right you have your base

function in this here and you have these other two so this second one that's y = absolute x - 3 as we saw in the table

the Y values were all three lower and in this situation the graph is 3 units lower every single point this vertex here is three units lower uh this point

is three units lower every single point is three units lower on that graph right makes sense because if you subtract three in your calculation process your y values are all three lower and the same

goes for the second one there if we look at that one this y = Absol 5x + 4 all of those y values are four higher right every single one again this vertex is

four higher but every point on the curve is gone up by four because in your calculation process you worked out the absolute value then you added four so makes sense that it's four higher than

that so if you're going to describe this here what's happened in each of those cases we have the blue graph y =

absolute of x - 3 y = absolute of x - 3 has been you say translated down three

units and the other one then as you can probably predict here y = absolute value of x + 4 it has been

translated up four units and if you want a bit of an abbreviation here what I tend to call this is I call this one a vertical

translation bt3 down and this one up here call it a vertical translation for

up we're going to look now at horizontal translations and for this we're going to change it up and use a different base function y = x^2 that basic Parabola that you probably have seen before so

we're going to look at that equation and we're going to also change it to Y = X + 3 all 2 and Y = X - 4 All all squ in the

intro I talked about this as changing X to x + 3 replacing it or replacing x with x - 4 and we'll see what happens in

these tables here now with these tables you notice that these X values are different and in a second you'll see why that is but first we'll generate the Y

values and we're doing that just by substituting these values in for X in each of those all right so when you do that in the

interest of time here you get those values right there and as I said the X values are all different and why this is is because to really see what's going on

here it's helpful in this situation if we choose X values that are going to give us the same y values so these different X values

here are such that we always get these same y values here and to see that so if you've seen this function before you know that it's just X values and they

squares the Y values are just the squares of the x value so 0 squar is 0 1 squar is 1 2 s is 4 3 squ is 9 and so on right it would continue pass there and

the same goes the other way because when you square negatives it becomes positive now in the second case here what we're going to do is we're going to add three

before we Square this thing we're adding three before we square that thing if we want to achieve the same yalue which was my goal here if we want to achieve the

same yvalue if I'm going to add three I have to start with X values that are three lower than what I had over here

these values are all three lower than that and similarly for that second one if before I Square it I'm going to subtract four and I want to achieve

these same y values if the first thing I'm going to do is subtract four before I Square the number I have to start with X values that are all four higher than

these ones you see that these are all four more than those ones so just to summarize that there when you replace x

with x + 3 what happens is for the same y values the X values are all three lower if you replace it with x + 3 and

the X values are all four higher if you replace it with xus 4 so it's a little bit strange it works kind of the opposite you add something in here it makes these lower you subtract something

in there it makes these higher you may have seen that before too actually so we'll summarize it over here to get the same yv

values in that first one X values are three lower three less and in the

last function there the X values are four greater or more we'll just put here now we're going to graph these

three functions as well you can go ahead and graph the points from those tables of values and again in the interest of time those are the graphs that you're going

to get for each of them and as you see on the graph what has happened is a horizontal translation this function here Y = X + 3 all 2ar you noticed in

the table that the X values were all three lower for the same y values so what that means is for this same yv value of zero the x value is three lower

three to the left and this one where that X has been changed to x - 4 this x value is four greater in order to achieve that same y value of Z zero and

the same is true for all of these other ones every single point in this every single point on the graph moves it's not just that vertex that moves it's every single point right and on this every

single point here moves four to the right this point here four to the right and so on so what's happened there is that

this Y = X + 3 all sared has been translated three unit

left and the other one of course then has been this one y equal x - 4^ 2ar has

been translated four units right and again if you want some abbreviations there I would call this a

horizontal translation three left you can put an L or left like this if you if you want and then this one is horizontal translation for units

right so we're now going to try and take some of those Concepts and apply them here to describe what change has happened in a graph and write an equation for that so in this first one

here you notice that this is f ofx now we're trying to write an equation for this this is the transformed function how do you get G ofx from F ofx so you

take F of X if you look at it just pick any point on the graph it could be could be this one here that point the corresponding point on this graph is

right there or the corresponding point to that is right there and so on right this one up here is right there if you look at that that is five units to the

left so that that change is five units to the left so if you're trying to describe this you're going to say that this is a horizontal translation five

units left of using my abbreviations and if you're trying to write an equation for that then the equation that you can write for the blue

graph there is you can say that g ofx an equation for G ofx is if you take F ofx now to shift it five units to the left

instead of having this X in here we're going to replace that x with x + 5 because if we put x + 5

it goes in the negative Direction there so G ofx is equal to F ofx + 5 in this second one here you notice that every

point on this graph f ofx is shifted down looks like six units there to get the graph of G ofx this has gone down six units any point here every point on

the graph I'm not going to draw them all but all those points are shifted down six units and so you're going to say that this has had a vertical translation of six units

down and if you want to write the equation of it to get G of X you take F ofx now to get F ofx to shift down six

units you're going to put a Min - 6 outside of the function here when you make that change outside of the function it's going to change the Y values so

those are those two equations for that we're next going to look at a situation where we have a given function

this graph here and we're going to graph this transformed function here so first of all we need to recognize what those changes represent what they're going to

cause so if you change X to xus 2 that is going to cause a horizontal translation of two units to the right

because when you replace it with xus 2 does the opposite the X values have to be to Greater in our transformed function and then we

also have this + five on the end and that is going to cause a vertical translation and this does what you think it does it goes five up when it's plus five when it changes y it Chang changes

it the way you think it's going to change it plus five increases the value and the horizontal one's the one that does kind of the opposite of what you think so to be able to draw the graph

all you do is do those two changes to each of the points now you can just work with these sort of key points along here and transform each of those so if you

take this point and you go two to the right and five units up it's going to end up there and the same goes for each of these other ones two to the right

five up goes to there and this one same thing you're going to find it's right there and those last two two to the

right right five up and lastly this one two to the right five up it's there so when you're drawing this graph of this transformed function now I will try to draw the

straightest lines I can here but the computer doesn't always tend to do that when you're using the pen it makes them a little Wiggly that's not too bad and

connect those ones connect those ones that is quite Wiggly let's do that one again not any better I don't think there you go so that's that

function and we can label it we can call it y = f of x - 2 + 5 here we have a little bit more

complicated looking function that we've had before f ofx is 3x^2 - 2x + 1 and we're going to write the equation of the transformed function after two

translations have happened one that is three units to the left and and one of that is seven units down now the first thing that we need to do here is establish what it is we need to do to the equation to get those two things to

happen now this first one here to get that to happen we need to take our X and we need to replace it with x + 3 to go in the negative Direction the left we

have to add 3 to X there and to get it to go seven units down we need to subtract s from the function so to make

things a little bit easier here I'm going to use the notation G ofx to represent the transformed function so to

do that I'm going to take F ofx but instead of X in here I am going to replace that with x + 3 and then on the end of the function I'm going to

subtract 7 so that's what G of X is there so I can write an equation out using that specific function this specific function but instead of writing

3 x^2 I am going to put in here x + 3 and instead of writing - 2 X I am going to

write instead of that X I am going to put x + 3 x appears in two places in this original function you have to replace both of them to get this change

to happen and then of course I had the plus one and to get the vertical change to happen I have to subtract seven from the end of that function so that

function is the transformed function but it probably helps if I simplify it a little bit so we're going to write that g of x if you expand this x + 3 all SAR

before you multiply the three in so I'm going to leave the three there expand this x + 3^ 2ar so x + 3 * x + 3 is x^2

+ 6 x + 9 and I will leave this one the way it is for now we can expand both of those together and then I can actually

combine these two right now + 1 - 7 is -6 and if we keep going here now we can distribute

this there and so I'm going to have 3x² + 18x + 27 and if I distribute this

I am going to get - 2x but -2 * + 3 is - 6 and then of course I have this - six still there if I combine all my like

terms to make it as simple as possible there's only one x s term but there are two x terms so plus 18x and - 2x is + 16

x and then I have three terms that can go together the + 27 - 6 - 6 27 - 6 - 6 is 15 so that's the equation of the

transform function in simplest form now just look looking at the equation it's hard to tell that that's what's happened once you've simplified it but if you wanted to check to see that this is

indeed the graph you get after this function if you have graphing technology you can put this one in and this one in and just confirm that the graphs are

apart by three and seven so in summary here about translations when you have a function f ofx and you change it to Y = FX + K the

graph undergoes a vertical translation and that vertical translation is up if K is greater than zero or in other words

if it's a positive number and it gets translated down if K is less than zero or in other words if

it's a negative number and then when you have F ofx and it's changed to f ofx plus H or in other

words if you replace the X with x + H that graph undergoes a horizontal translation and the horizontal

translation is to the right if H is less than zero or in other words if it's

negative and it is to the left if H is greater than zero or in other words if H is positive so this is the one where it

works the opposite of what might think when H is negative it gets shifted in the positive direction and when H is

positive it gets shifted in the negative Direction and then lastly here translations of a function result in

graphs where the shape and orientation of the original remain unchanged the graph has shifted but the shape and

orientation have not changed at all that's [Music] [Music] it