Understanding Function Transformations in Algebra

Good day students! Welcome to mathcodeserved.com. In this clip, we're going to be taking a look at transformation of functions. To get us started, we're going to be taking a look at the formulas that will be guiding our problem solving process. So we're going to go to our website, mathcodeserved.com, and then on the Algebra 2, we're going to go to our formula collection. And if you take a look at the most known Algebra 2 formulas, the first formula that we're looking at, Transformation Rules for Functions, these are the functions that um the formulas that we're going to be using for our examples today. All right, we're going to focus our attention mainly on the first four, basically vertical and horizontal shifts. All right, so the instructions are as follows. So for the given functions, for the given functions, we're to do the following. Okay, for the given functions. Identify the family. Identify the family that the function belongs to. The shifts or the transformations. After doing that, you're to sketch the graph. Okay, sketch the graph of the function. Alright, so. To get us started, let's take a look at the function y equals the absolute value of x plus 2 as a first transformed function. Alright, so what family does this function belong to? Let's go ahead and identify it. So the family that this function belongs to is the absolute value family. Absolute value. Alright, this function looks like a V. If you want to describe how it looks like using a letter, it looks like like a V and the transformational form, the formula for it is y equals a times the absolute value of x minus h plus k. All right. OK, so next, let's take a look at the transformations for this. particular function. So what kind of transformations are happening to the parent function? Well, let's take a look at the formula, the transformational form right here. What we're going to do is we're going to proceed to write the given function in the transformational form. If we do that, we're going to have y equals, now the invisible coefficient in front of this absolute value bracket is 1, so y equals 1. Whenever you don't have a coefficient, there's only 1. times x plus 2, all right? And you're adding nothing here, so you're just adding 0, okay? So don't forget that the number next to the x tells you the shifts left or right, and the number that's outside the function tells you the shifts up or down, all right? This is your h left, right, k up, down. Now, if you notice in the formula, the h is negative, there's negative h. That means you're going to be doing the opposite move, all right? So since you have plus 2 and you have a negative h, that basically means you have minus minus 2, right? So that means that we're going to be shifting in the left direction. So x plus 2, we're going to shift 2 units to the left. like that all right let's go ahead and graph the function so what we're going to do is we're going to sketch the parent um absolute value function using dotted lines and then we'll apply the transformation okay all right so let's go ahead and sketch the absolute value function so to do that we're going to need our points our guiding points let's put it right here So these are the points that are going to be guiding our construction. We don't need too much. All right. So X and Y, fix that, X and then Y equals the absolute value of X. That's the parent function. We're going to just use negative 1, 0, and 1 for the input values output. Absolute value of negative 1 is just 1. Absolute value of 0 is 0. And absolute value of 1. is one. Alright, so these are the guiding points, the base points that we're going to use to generate our parent function. So we have negative one. negative 1 1 here 0 0 and then 1 1 all right let's go ahead and graph our parent function let's make it slimmer all right so our parent function let's put it in green it's something like this so this is the original this is the absolute value of x function it's the v function all right so now what we're going to do is we're going to go ahead and apply the transformation all right we're going to apply the transformation rule that we indicated earlier let's make sure this points around there right here right here bam bam bam so what we're going to do is simply move these points two units to the left for our shift here okay so you see this vertex right here we'll move it two to the left one two let's make it blue one two takes it right here and then this point move it to the left one two take that right here and then this point to the right one two takes you right here you have an overlapping situation there okay so what we're gonna do is we're gonna go ahead and uh draw our final function the shifted function transforms function there you go right there bam bam all right so there goes your final answer this function that we just drew the black one is y equals the absolute value of x plus 2. Okay? All right, let's move on to the next example, number 2. Let's say we have the function y equals x squared minus 4 as the example. Okay, so what family does this function belong to? Let's go ahead and specify that. So the family that this function belongs to is the quadratic family. Why? Because you have a square right there. The highest degree for a variable component is a square. The polynomial of second degree is quadratic. Just like here we have the absolute value bars. So that means it's absolute value. Okay. All right. What does the quadratic function look like? The quadratic family still look like u's. The absolute value look like v's. Quadratic look like u's. Okay, and then the transformational form is y equals a times x minus h quantity squared plus k. Bam, bam, bam. Okay, so let's find what the transformations are for this function. So we're just going to write down the given function in its transformational form. If we do that, we're going to have... y equals, there's no number next to the x, right? The function, so we'll put a 1. That's the visible coefficient. x, now what are you subtracting from the x that's being squared? If you group this like this, nothing is being subtracted from the x, right? So nothing is 0, so you subtract nothing from the x. Square minus 4, okay? So you notice that we just have a shift in the vertical direction, up or down. That's for the k component. does. All right. The X is left, right. The K is up, down. So let's indicate it here. So we're going to shift. We're going to shift four units down. That's what's going to happen. Let's go ahead and graph our function. All right. So you're just going to have a vertical shift of four units. downwards. So to get us started, we are going to generate our guiding points. It's going to use three pair of coordinates for this. So our guiding points are as follows. X and then the absolute value of, I'm sorry, YX squared. is the output for the parent. Okay, so negative 1, 0, 1. If you square negative 1, negative 1 times negative 1 is positive 1. 0 times 0 is 0. 1 times 1 is 1. So your guiding points are here. 0, 0, 1, 1. And negative 1, 1. Alright, so remember what this function looks like. The quadratic family is like a U. all right so we're going to connect these dots using a u function or also called the parabola and it's going to be upward opening okay so let's make it green so our parent function the original will look something like this let's try it again so something like this just like a u opening up and then this one also upward opening parabola okay so what we're going to do now is we're going to shift it four units down and we shifted four units down just take these three points and ship them four units down one two three the vertex ends up right there this point on the right one two three four and this point on the left one two three four okay and then just connect the dots your upward opening parabola solid line because that's the Final answer. So that part again. Okay, so there you have it. So this graph right here is the final answer. This is y equals x squared minus 4. Okay, let's move on to the next one. Question number 3. Let's say we have y equals the square root. Of x minus 4. Alright, what family does this function belong to? We have a square root, so that tells you it belongs to the radical family. Okay? So, what does the radical family look like? The graph kind of looks like this right here. Starting from 0 and then just goes up. Alright? The transformational form is given by y equals a times the square root of x minus h plus k. Bam, bam, bam. All right, now let's talk about the transformations. Transformations. To determine what the transformations are, we're going to just write down the parent function. in transformational form. If we do that, we are going to have y equals, there's no coefficient, right? So you have one times the square root of x minus h, no, x minus four, one times x minus four, all right, equals that square root. And then There's not even an added on, so just add 0. They have it, okay? So what are your transformations? Your transformations here are just horizontal. Remember, the number next to the x specifies a horizontal shift. So here we're going to shift 4 units. Remember, since you have a minus here next to the x, it indicates opposite movement, right? 4 units to the right. four units to the right and that's it let's go ahead and graph this uh so let's create our coordinate system all right we're going to need some points to help us uh carry out our parent graph construction and then and then the resulting transformation. Okay, so let's put the points in. So our points are as follows. We have x, and then y is going to be the square root of x. For our input values, we're going to just pick 0. 1 is a perfect square root of 4. All right, so the square root of 0 is 0. The square root of 1 is 1. The square root of 4 is 2. So 0, 1, and 2. Let's go ahead and graph those points. Graph those points. We have 0, 0, 1, 1. And then for 4, 1, 2, 3, 4. Right there. Bam. Okay. So if we draw the parent function, the original parent function, it's going to be something like this. So this one is y equals square root of x. okay let me just specify what these are y equals x square and then this one is y equals absolute value of x okay so we're going to shift it here just four units to the right so just shift the points four units to the right So if you do that, you're going to get this one goes to the right. One, two, three, four. And then this one, one, two, three, four. And then this one right here, one, two, three, four. Running into my labels. Okay. So all we just do is just connect it with a solid line right there. We're going to connect it using a solid line because that is our final result. Okay, so this one is y equals root x. Okay, so if we connect the lines, we're going to have something like this through there, through there. Okay, and that's it. This is y equals the square root of x minus 4. Alrighty, let's take a look at one more example. Okay, so what if we have the given function? Let's carry out two transformations now, okay? Number four, what if we have y equals the quantity x minus three cubed plus two, okay? All right, let's find out what a family is. A family, since the highest power of this polynomial function is three, is a cubic family, okay? What does a cubic family look like? It's kind of like this. Let's do that again. Something like that. Alright, and then the transformational form for the cubic family is given by y equals a times x minus h quantity cubed plus k. Alright, so this will help us find what the transformations are. So the transformations are as follows. So let's write down the original parent function in transformational form. So that's going to be y equals the number in front of this quantity cubed. So for the 1, 1 times x minus 3 quantity cubed plus 2. All right, so we can clearly see what our shifts are. So you have minus 3, so we're going to be going to the right. And you have plus 2, that means we're going to be going up. Okay. So our transformations are we're going to shift shift three units to the right and two units up. Okay, let's go ahead and generate the graph of the function. All right, so to generate the graph of the function, we're going to need some points to guide our construction. So let's put in the points. Points are as follows. So we're looking at x and y equals x to the third. There's a parent cubic function. All the points that we're going to be generating first, and then we'll go on after that. All right, so. We're going to use input values negative 1, 0, 1. These are the guiding points for this. All right, so when you cube negative 1, negative 1 times negative 1 times negative 1 is negative 1. 0 to the third is 0. 1 to the third is 1. So you have it like that. Okay, let's draw the points for the cubic function. So we have 0, 0, negative 1, negative 1. right there um one one right there bam bam bam connect the dots in the lights so it's gonna look something like this on the right side it's the it's upwards open concavity and then on the left side is downwards right so it's like that okay so that goes your uh parent cubic function so to um Generate the transform function you're just going to apply some transformations to it what you're going to do is shift it All the points you're going to shift them three units to the right and between it's up So one two three three to the right and two up one two That's the zero point right there Okay, and then from here you can just figure out the other two because one is on upper right and the other is on the lower left Draw the lines using solid lines. This is our final answer. Bam. And this one, try to make it symmetric, right? To an extent. All right. So there you have it. So this line that we just drew right here is y equals quantity x minus 3 to the third plus 2. OK, so that's that. Thanks so much for taking the time to watch this presentation. A special thank you for all those who've been supporting our channel with your likes and comments and subscriptions. If you have any questions about this tutorial just specify in the comment section below. Tons of support resources can be found on macgutserved.com. Thanks again for watching and have a wonderful day. Goodbye.

And if you take a look at the most known Algebra 2 formulas, the first formula that we're looking at, Transformation Rules for Functions, these are the functions that um the formulas that we're going to be using for our examples today. All right, we're going to focus our attention mainly on the first four, basically vertical and horizontal shifts. All right, so the instructions are as follows. So for the given functions, for the given functions, we're to do the following.

Okay, for the given functions. Identify the family. Identify the family that the function belongs to.

The shifts or the transformations. After doing that, you're to sketch the graph. Okay, sketch the graph of the function. Alright, so. To get us started, let's take a look at the function y equals the absolute value of x plus 2 as a first transformed function.

Alright, so what family does this function belong to? Let's go ahead and identify it. So the family that this function belongs to is the absolute value family. Absolute value. Alright, this function looks like a V. If you want to describe how it looks like using a letter, it looks like like a V and the transformational form, the formula for it is y equals a times the absolute value of x minus h plus k.

All right. OK, so next, let's take a look at the transformations for this. particular function.

So what kind of transformations are happening to the parent function? Well, let's take a look at the formula, the transformational form right here. What we're going to do is we're going to proceed to write the given function in the transformational form. If we do that, we're going to have y equals, now the invisible coefficient in front of this absolute value bracket is 1, so y equals 1. Whenever you don't have a coefficient, there's only 1. times x plus 2, all right? And you're adding nothing here, so you're just adding 0, okay?

So don't forget that the number next to the x tells you the shifts left or right, and the number that's outside the function tells you the shifts up or down, all right? This is your h left, right, k up, down. Now, if you notice in the formula, the h is negative, there's negative h. That means you're going to be doing the opposite move, all right?

So since you have plus 2 and you have a negative h, that basically means you have minus minus 2, right? So that means that we're going to be shifting in the left direction. So x plus 2, we're going to shift 2 units to the left.

like that all right let's go ahead and graph the function so what we're going to do is we're going to sketch the parent um absolute value function using dotted lines and then we'll apply the transformation okay all right so let's go ahead and sketch the absolute value function so to do that we're going to need our points our guiding points let's put it right here So these are the points that are going to be guiding our construction. We don't need too much. All right.

So X and Y, fix that, X and then Y equals the absolute value of X. That's the parent function. We're going to just use negative 1, 0, and 1 for the input values output.

Absolute value of negative 1 is just 1. Absolute value of 0 is 0. And absolute value of 1. is one. Alright, so these are the guiding points, the base points that we're going to use to generate our parent function. So we have negative one.

negative 1 1 here 0 0 and then 1 1 all right let's go ahead and graph our parent function let's make it slimmer all right so our parent function let's put it in green it's something like this so this is the original this is the absolute value of x function it's the v function all right so now what we're going to do is we're going to go ahead and apply the transformation all right we're going to apply the transformation rule that we indicated earlier let's make sure this points around there right here right here bam bam bam so what we're going to do is simply move these points two units to the left for our shift here okay so you see this vertex right here we'll move it two to the left one two let's make it blue one two takes it right here and then this point move it to the left one two take that right here and then this point to the right one two takes you right here you have an overlapping situation there okay so what we're gonna do is we're gonna go ahead and uh draw our final function the shifted function transforms function there you go right there bam bam all right so there goes your final answer this function that we just drew the black one is y equals the absolute value of x plus 2. Okay? All right, let's move on to the next example, number 2. Let's say we have the function y equals x squared minus 4 as the example. Okay, so what family does this function belong to?

Let's go ahead and specify that. So the family that this function belongs to is the quadratic family. Why?

Because you have a square right there. The highest degree for a variable component is a square. The polynomial of second degree is quadratic. Just like here we have the absolute value bars. So that means it's absolute value.

Okay. All right. What does the quadratic function look like? The quadratic family still look like u's.

The absolute value look like v's. Quadratic look like u's. Okay, and then the transformational form is y equals a times x minus h quantity squared plus k.

Bam, bam, bam. Okay, so let's find what the transformations are for this function. So we're just going to write down the given function in its transformational form. If we do that, we're going to have...

y equals, there's no number next to the x, right? The function, so we'll put a 1. That's the visible coefficient. x, now what are you subtracting from the x that's being squared? If you group this like this, nothing is being subtracted from the x, right? So nothing is 0, so you subtract nothing from the x.

Square minus 4, okay? So you notice that we just have a shift in the vertical direction, up or down. That's for the k component. does. All right.

The X is left, right. The K is up, down. So let's indicate it here.

So we're going to shift. We're going to shift four units down. That's what's going to happen. Let's go ahead and graph our function.

All right. So you're just going to have a vertical shift of four units. downwards. So to get us started, we are going to generate our guiding points. It's going to use three pair of coordinates for this.

So our guiding points are as follows. X and then the absolute value of, I'm sorry, YX squared. is the output for the parent.

Okay, so negative 1, 0, 1. If you square negative 1, negative 1 times negative 1 is positive 1. 0 times 0 is 0. 1 times 1 is 1. So your guiding points are here. 0, 0, 1, 1. And negative 1, 1. Alright, so remember what this function looks like. The quadratic family is like a U.

all right so we're going to connect these dots using a u function or also called the parabola and it's going to be upward opening okay so let's make it green so our parent function the original will look something like this let's try it again so something like this just like a u opening up and then this one also upward opening parabola okay so what we're going to do now is we're going to shift it four units down and we shifted four units down just take these three points and ship them four units down one two three the vertex ends up right there this point on the right one two three four and this point on the left one two three four okay and then just connect the dots your upward opening parabola solid line because that's the Final answer. So that part again. Okay, so there you have it. So this graph right here is the final answer.

This is y equals x squared minus 4. Okay, let's move on to the next one. Question number 3. Let's say we have y equals the square root. Of x minus 4. Alright, what family does this function belong to?

We have a square root, so that tells you it belongs to the radical family. Okay? So, what does the radical family look like?

The graph kind of looks like this right here. Starting from 0 and then just goes up. Alright? The transformational form is given by y equals a times the square root of x minus h plus k.

Bam, bam, bam. All right, now let's talk about the transformations. Transformations.

To determine what the transformations are, we're going to just write down the parent function. in transformational form. If we do that, we are going to have y equals, there's no coefficient, right? So you have one times the square root of x minus h, no, x minus four, one times x minus four, all right, equals that square root.

And then There's not even an added on, so just add 0. They have it, okay? So what are your transformations? Your transformations here are just horizontal. Remember, the number next to the x specifies a horizontal shift. So here we're going to shift 4 units.

Remember, since you have a minus here next to the x, it indicates opposite movement, right? 4 units to the right. four units to the right and that's it let's go ahead and graph this uh so let's create our coordinate system all right we're going to need some points to help us uh carry out our parent graph construction and then and then the resulting transformation.

Okay, so let's put the points in. So our points are as follows. We have x, and then y is going to be the square root of x.

For our input values, we're going to just pick 0. 1 is a perfect square root of 4. All right, so the square root of 0 is 0. The square root of 1 is 1. The square root of 4 is 2. So 0, 1, and 2. Let's go ahead and graph those points. Graph those points. We have 0, 0, 1, 1. And then for 4, 1, 2, 3, 4. Right there. Bam. Okay.

So if we draw the parent function, the original parent function, it's going to be something like this. So this one is y equals square root of x. okay let me just specify what these are y equals x square and then this one is y equals absolute value of x okay so we're going to shift it here just four units to the right so just shift the points four units to the right So if you do that, you're going to get this one goes to the right. One, two, three, four.

And then this one, one, two, three, four. And then this one right here, one, two, three, four. Running into my labels. Okay.

So all we just do is just connect it with a solid line right there. We're going to connect it using a solid line because that is our final result. Okay, so this one is y equals root x. Okay, so if we connect the lines, we're going to have something like this through there, through there. Okay, and that's it.

This is y equals the square root of x minus 4. Alrighty, let's take a look at one more example. Okay, so what if we have the given function? Let's carry out two transformations now, okay?

Number four, what if we have y equals the quantity x minus three cubed plus two, okay? All right, let's find out what a family is. A family, since the highest power of this polynomial function is three, is a cubic family, okay? What does a cubic family look like? It's kind of like this.

Let's do that again. Something like that. Alright, and then the transformational form for the cubic family is given by y equals a times x minus h quantity cubed plus k. Alright, so this will help us find what the transformations are.

So the transformations are as follows. So let's write down the original parent function in transformational form. So that's going to be y equals the number in front of this quantity cubed. So for the 1, 1 times x minus 3 quantity cubed plus 2. All right, so we can clearly see what our shifts are. So you have minus 3, so we're going to be going to the right.

And you have plus 2, that means we're going to be going up. Okay. So our transformations are we're going to shift shift three units to the right and two units up.

Okay, let's go ahead and generate the graph of the function. All right, so to generate the graph of the function, we're going to need some points to guide our construction. So let's put in the points.

Points are as follows. So we're looking at x and y equals x to the third. There's a parent cubic function.

All the points that we're going to be generating first, and then we'll go on after that. All right, so. We're going to use input values negative 1, 0, 1. These are the guiding points for this. All right, so when you cube negative 1, negative 1 times negative 1 times negative 1 is negative 1. 0 to the third is 0. 1 to the third is 1. So you have it like that. Okay, let's draw the points for the cubic function.

So we have 0, 0, negative 1, negative 1. right there um one one right there bam bam bam connect the dots in the lights so it's gonna look something like this on the right side it's the it's upwards open concavity and then on the left side is downwards right so it's like that okay so that goes your uh parent cubic function so to um Generate the transform function you're just going to apply some transformations to it what you're going to do is shift it All the points you're going to shift them three units to the right and between it's up So one two three three to the right and two up one two That's the zero point right there Okay, and then from here you can just figure out the other two because one is on upper right and the other is on the lower left Draw the lines using solid lines. This is our final answer. Bam.

And this one, try to make it symmetric, right? To an extent. All right.

So there you have it. So this line that we just drew right here is y equals quantity x minus 3 to the third plus 2. OK, so that's that. Thanks so much for taking the time to watch this presentation.

A special thank you for all those who've been supporting our channel with your likes and comments and subscriptions. If you have any questions about this tutorial just specify in the comment section below. Tons of support resources can be found on macgutserved.com. Thanks again for watching and have a wonderful day. Goodbye.

Transcript for:Understanding Function Transformations in Algebra

Transcript for:
Understanding Function Transformations in Algebra