welcome to lesson 2.2.8 linear functions in Transformations our objectives today are to graph linear equations using Transformations which means you have to be able to identify transformations in the equation so let's look at our parent function our parent function is yals X it's very basic um there is no transformation in this equation at this time what we need to notice though our few of a couple characteristics that are important to point out in a parent function and the first one is the Y intercept notice there is nothing on the end um normally you guys are used to thinking of yal mx + b a linear equation the B is your Y intercept in this case b is zero so our Y intercept is 0 0 also in this pair and function there is no number in front of the X which means it has a slope of one or one over one and you can see that in the graph because it goes right through 0 0 the Y intercept and if I count up one and over one that's where my next point is that's my slope and I could continue to do that remember that's one of the characteristics of a linear equation is that the slope has a constant rate of change it's always the same as you move up now let's look at the transformations of the parent function of a linear parent function so we get away from M being slope and now we call this a and you'll see as we move on through different types of functions that we tend to use a b c d as the Transformations within the function so we will call the transformation to the slope a and we will call the transformation to the Y intercept B there's a third transformation in a linear function and you can't see it in here because right now we have a positive but could have a negative in front of the slope and if you did have a negative in front of the a that transformation is called a reflection and you'll see that when we graph the equation the equation actually reflects it reflects over the um Y axis the second one here the a if we had something other than one then we would have a transformation that's going to be our change in our slope so I've listed down here and again if you're taking notes make sure you write this down it's going to be very important and it will remain consistent no matter what type of function we're setting so these nodes can stay with you for the rest of the year if you have a slope that is greater than one so a slope of two a slope of three even a slope of three halves because that's greater than one that's consider a stretch the line will get steeper it will stretch up on the other hand if you you have a slope that's less than one say/ 12 or 1/3 or 1/4 or even 2/3 that's small that means that you have a shrink in fact your line is getting less steep it is shrinking back down to the x axis think of the x-axis as the ground it shrinks back down to the ground the third transformation with a linear function is your B value and that's a shift it's either a shift up or a shift down so if you have something other than zero say it's a plus two or a plus three in other words if it's positive your whole line will shift up and it makes sense because your Y intercept will no longer be0 0 it'll be whatever is in the B value if the B value is negative say you have a -4 -10 your whole line will shift down let's look at some of these examples actually so in this example we have the equation Y = X + 5 in this equation you can see the parent function here in purple going right through 0 with a slope of up one over one but when we transform it by having a shift up a five right here on the end our line just picks up and shifts upwards oops let's go back shifts upward to five right here nothing has changed in the slope you'll notice our slope has no transformation all that we've done is we've taken our Y intercept and we've shifted it now up to five here's another example if we look at our transformation on this one we have a minus three well we learned that that would be a shift down of three and again here's our parent function in purple if you'll notice our transformation says that we shift down here to three but our slope stayed the same we did not have a transformation to the slope so now let's look at some transformations to our slope in this case we have a transformation of four remember that anything greater than one is going to be a stretch so now we have a stretch of four and again you can compare the purple which is our parent graph to this brown one you can see the brown one is much steeper because now we have stretched up four we have no transformation to the Y intercept so it still goes through 0 0 but let's count together so from here if we go up 1 2 3 4 and then run over one there's the next point and that comes from our four let's look at another one in this one we have a slope of 1 over three that's less than one so we have a shrink of 1/3 and again let's compare the purple graph is our parent function goes right through 0 0 up one over one our slope is up one over one but now we've changed our slope because we have a transformation instead we're going to go up one and over 1 2 3 right here on the brown graph that gives us a transformation of 1/3 which is a shrink of 1/3 it is less steep than our parent function it is shrinking down okay let's try one more in this case now we do have a negative in front of the X X in other words in front of a a is understood to be one remember that this kind of transformation is called a reflection and now you can clearly see how our purple line which is our parent function has reflected itself over this y AIS and it looks exactly the same but on the other side if you were to put some sort of mirror here in the middle the brown one would be just be a reflection of the purple one okay and that is a separate transformation all and of itself it's called a reflection let's start putting some together so say you have the graph of y = -5x we have two Transformations because we have the negative which we said as a reflection we have the five which is greater than one so we have a stretch of five and when we graph that we can see that it's the brown graph so it has been reflected over the y axis and it's a lot steeper because it has a stretch of five now let's look at another one this one has a slope of2 and a y intercept of four so when we identify those as Transformations we say that it has a shrink of2 because it's less than one meaning the graph will become less Steep and you can see the brown graph is less steep than our parent function which is the purple graph the B value which is four means that the whole entire graph shifts up from 0 0 which is our parent function so here it is shifted up to four so we have two transformations in this linear function okay let's look at another one in this case we have three which is a stretch and we have a -6 on the end which is a shift down so we have a stretch of three and a shift down of six starting from our parent function here if we shift down down to six there's our Y intercept and then if we look at our slope we went 1 2 3 over one and that made our graph much steeper let's do some examples so in your practice you will have to you will be given a linear equation and you will have to list the transformations to the parent function and then identify a graph or graph the function so if we look at this one y = x - 8 the only transformation on here is the minus 8 remember that yal X is a parent function anything that is added on to that is in fact a transformation well if you remember the minus 8 is a shift down shift down of eight that's the transformation in order to graph this we would first plot our Y intercept of this 08 then we need a second point to draw a line so we always go to our slope well on this one we have no transformation it's just 1 over one so we go up one and over one and put our second point there and that's how we would get our line okay number two in this example we have two Transformations we have this two on the end then we have a four in front of the X the four is our stretch or Shrink and since it's greater than one it's a stretch of four the two at the very end that's our B value and that's a shift up of two oops that should say two when we graph this we would plot the Y intercept at 02 for the two on the end which is right there and then we would plot our second Point using our slope here of four which is over one so we would go up 1 2 3 4 and over one and there is our second Point let's try one more in this example we have all three Transformations we have the negative out front we have 1 12 adjusting our a value or slope and then we have a five adjusting our B value or our Y intercept so first things first the negative means we have a reflection the 1/2 means we have a shrink because it's less than one so we say a shrink of 1/2 and the five means we shift up five it's positive five a reflection a shrink of 1/2 and a shift up of five now when we're graphing this we graph it the same we always go to our Y intercept first and we plot this point then we look at our slope in this case let's take the negative and the 1/2 together since it's a negative slope and we always read our graphs from left to right so we would start on the left side of the graph we would need to make sure that the line angles down we're going down slope if it's negative so from here we have rise over run and our rise is one and our run is two so we can either go up one and over two here or we can go down one as long as our rise is up or down and over two to here and plot our point and that's how we graph using Transformations and these will help us in the future to graph more complicated functions so it's good to learn them on linear functions because they're more simple and then we'll progress using more complicated functions so just to wrap up remember that we have three three Transformations we have Reflections which is a negative we have stretches or shrinks which pertain to our slope and then we have shifts up or down which is the Y intercept and that concludes this lesson