Video: Transforming Functions: What Happens to the Equations?

Video Transcript

In this series of videos, we’ve looked at what happens to the graphs of functions when you transform them in different ways. We’ve seen vertical and horizontal translations and vertical and horizontal stretches. Now, we’re gonna take a few minutes to look at what happens to the equation of a function when we transform it.

We’ve seen that 𝑓 of 𝑥 plus 𝑎 is a transformation that translates 𝑓 of 𝑥 by negative 𝑎 units in the 𝑥-direction. So, if 𝑎 is positive, it’s gonna translate the curve of the function to the left. And if 𝑎 was negative, it’s gonna translate the curve of the function to the right. We’ve also seen that 𝑓 of 𝑏 times 𝑥 stretches 𝑓 of 𝑥 by a factor of one over 𝑏 in the 𝑥-direction. And that fixes the curve at the 𝑦-axis and stretches the curve away from the 𝑦-axis. Although depending on the value of 𝑏, it might squash the curve towards the 𝑦-axis.

And we’ve also seen that 𝑓 of 𝑥 plus 𝑎 translates the function 𝑓 of 𝑥 by 𝑎 units in the 𝑦-direction. So, if 𝑎 is positive, it shifts the curve upwards. And if 𝑎 is negative, it shifts the curve downwards. And we’ve seen that 𝑏 times 𝑓 of 𝑥 stretches the function 𝑓 of 𝑥 by a factor of 𝑏 in the 𝑦-direction. Now, in this case, it’s gonna lock off all points on the 𝑥-axis. And it’s either gonna stretch that curve away from the 𝑥-axis or it’s gonna squash it towards the 𝑥-axis. There’re various different things that can happen for different values of 𝑏.

Now, let’s think about some equations and see the effect of these transformations on them. First, let’s think about 𝑓 of 𝑥 plus 𝑎. So, for example, if 𝑓 of 𝑥 was three 𝑥 squared plus two 𝑥 plus one and 𝑎 was equal to five. Then as we said in the introduction, the curve of 𝑦 equals 𝑓 of 𝑥 is translated five places in the positive 𝑦-direction. So, take any point on the 𝑦 equals 𝑓 of 𝑥 curve, add five to its 𝑦-coordinate, and you’re gonna map onto the 𝑦 equals 𝑓 of 𝑥 plus five curve.

So, that’s the graph, but let’s think of the equation of 𝑓 of 𝑥 plus five. Well, we’re gonna start off with the expression for 𝑓 of 𝑥, so three 𝑥 squared plus two 𝑥 plus one. And then, we’re gonna be adding five to that, which simplifies to three 𝑥 squared plus two 𝑥 plus six. Now, in this format here, it’s pretty easy to see that we’ve taken the 𝑦-coordinates that we had from the 𝑓 of 𝑥 function and we’ve just added five to it. But in this format here, it’s perhaps not quite so immediately obvious what’s happened in terms of that transformation, even though it’s not a very dramatic change to the equation.

Now, let’s think of transformations of the type 𝑏 times 𝑓 of 𝑥. For example, 𝑓 of 𝑥 again, we’re gonna take three 𝑥 squared plus two 𝑥 plus one. And 𝑏 is gonna be equal to five. If we look at the curves of those two functions, 𝑦 equals 𝑓 of 𝑥 and 𝑦 equals five times 𝑓 of 𝑥, all we’ve done is we’ve taken the 𝑦-coordinates from 𝑦 equals 𝑓 of 𝑥 and we’ve multiplied them by three. Looking at the curves of those functions, we can see that the 𝑦-coordinates from 𝑦 equals 𝑓 of 𝑥 have just been multiplied by five to give us the 𝑦-coordinates of 𝑦 equals five times 𝑓 of 𝑥. So, one times five gives us five. Two times five gives us 10. Four times five gives us 20, and so on.

Now, looking at the equation of the function five times 𝑓 of 𝑥 is equal to five times three 𝑥 squared plus two 𝑥 plus one. And we can distribute that five across the parentheses and simplify the expression to five 𝑓 𝑥 is equal to 15𝑥 squared plus 10𝑥 plus five. Now, in this simplified format, it’s pretty much the same format, but all of the coefficients are five times larger. But it’s not immediately obvious what that transformation is doing.

So, if you look at the other format here, so we’ve got the same 𝑦-coordinates we have for 𝑓 of 𝑥, but we’ve just multiplied them all by five. So, just like before, by simplifying that expression, we’ve actually put it into a format which makes it harder to work out exactly what the transformation was doing, multiplying all of the 𝑦-coordinates by five.

Well, now, let’s have a look at transformations of the type 𝑓 of 𝑥 plus 𝑎. So again, we’re gonna use 𝑓 of 𝑥 is equal to three squared plus two 𝑥 plus one. And we’re gonna use 𝑎 equals five. And hopefully you remember from the introduction that a transformation like 𝑓 of 𝑥 plus 𝑎 translates the whole curve 𝑎 units to the left. So, because 𝑎 is equal to five, we’ve translated the 𝑓 of 𝑥 curve five units to the left.

Then, for example, this point on 𝑦 equals 𝑓 of 𝑥 maps over to here five places to the left on 𝑦 equals 𝑓 of 𝑥 plus five. And similarly, this point gets transformed to here. But it’s a little bit trickier when we try to think about what the function is going to look like in terms of the equation. In 𝑓 of 𝑥, we took 𝑥 and then we squared it and multiplied that by three. And then, we multiplied 𝑥 by two and then we added one. So, in this transformed function, wherever we saw 𝑥 in the original function, we’re gonna have to replace that with 𝑥 plus five.

So, instead of three 𝑥 squared, we’re gonna get three lots of 𝑥 plus five squared. And instead of two 𝑥, we’re gonna get two lots of 𝑥 plus five. But the plus one on the end isn’t affected, so that’s just gonna stay as plus one. So, now, we need to multiply out those expressions. And 𝑥 plus five all squared is 𝑥 squared plus 10𝑥 plus 25. And then, two times 𝑥 plus five, distributing the two across the parentheses there gives us two 𝑥 plus 10. Then, distributing that three through these parentheses here gives us three 𝑥 squared plus 30𝑥 plus 75. And then, just tidying up at the end, 10 plus one is 11.

So, the fully tidied up and simplified version 𝑓 of 𝑥 plus five is equal to three 𝑥 squared plus 32𝑥 plus 86 is still a quadratic. But it doesn’t look like an obvious simple change to transform it from the original function three 𝑥 squared plus two 𝑥 plus one. So again, if you left your transformed function in this unsimplified format, you might have a bit of a clue that what we were doing is translating the function five places to the left, where we’ve basically added five to all of the 𝑥-input values to the function. But if you’ve simplified into this format, then it’s not really an easy step to work out exactly what that transformation is going to be if you don’t know that we’ve called it 𝑓 of 𝑥 plus five.

So, lastly then, let’s consider function transformations of this format, 𝑓 of 𝑏 times 𝑥. Again, we’ll use the same 𝑓 of 𝑥 and 𝑏 will be equal to five. And remember that this type of transformation 𝑦 equals 𝑓 of five 𝑥 is gonna be a stretch times one over five in the 𝑥-direction about the 𝑦-axis. So, we’re looking off the 𝑦-axis. And all of the distances, all of the 𝑥-coordinates, are gonna be multiplied by a fifth.

If we started off with a 𝑥-coordinate of one, we’d multiply that by a fifth to make 0.2. And if our 𝑥-coordinate on 𝑓 of 𝑥 started off at negative two, we’d multiply that by a fifth, or divide by five, to get negative 0.4. Whereas any point which already had an 𝑥-coordinate of zero would map onto itself. So, we can see the whole thing is being squashed towards the 𝑦-axis.

So, what’s the equation of 𝑓 of five 𝑥 is going to look like? Well, again, 𝑓 of 𝑥 meant that we took the 𝑥-value, squared it, and multiplied by three. We took the 𝑥-value and doubled it, added that to the previous answer, and then added one. So, wherever we see 𝑥 in that function, in that expression there, we’re gonna replace it with five 𝑥. And this means that instead of getting three 𝑥 squared, we’re gonna get three times five 𝑥 all squared. And instead of just two times 𝑥, we’re gonna do two times five 𝑥. But of course, the plus one on the end is unaffected by all of this.

So, now, we’ve got to multiply this out. And five 𝑥 times five 𝑥 is 25𝑥 squared. So, we’ve got three times 25𝑥 squared. Two lots of five 𝑥 is 10𝑥, plus one. So, we can now tidy that up. So, in its simplified form, 𝑓 of five 𝑥 is 75𝑥 squared plus 10𝑥 plus one. So, again, it’s still a quadratic, but it’s not an easy and obvious simple change to transform that. So, again, in the simplified format, not obvious to work out what the transformation was. But if we left it in this unsimplified format over here, we can see that we’ve replaced the 𝑥s up here with five 𝑥s down here. And we can sort of work out what the transformation must have been.

So, I’ve shown you a method for transforming a function and putting it- simplifying it into a format which makes it difficult to work out what the transformation was. So, what’s the point of that? Well, you’re most likely to encounter it in questions like this.

Show that 𝑔 of 𝑥 is equal to 75𝑥 squared plus 10𝑥 plus one is a transformation on the function 𝑓 of 𝑥 is equal to three 𝑥 squared plus two 𝑥 plus one such that 𝑔 of 𝑥 is equal to 𝑓 of five of 𝑥.

So, what we’ve gotta do is we’ve gotta show that this expression here is this transformation of this expression here. So, let’s start off by saying 𝑓 of 𝑥 is equal to three 𝑥 squared plus two 𝑥 plus one. Then, we’re gonna form an expression for 𝑓 of five 𝑥 because that’s the transformation that the question says has happened. So, we’re gonna replace 𝑥 in the function with five 𝑥. And then, we can simplify it step-by-step. And this gives us the expression that they ask for in the question. And they told us that that was equal to 𝑔 of 𝑥. So, this was just an exercise in rearranging a formula and picking out bits from the question.

Let’s look at some more questions that involve manipulating the equations of functions.

𝑓 of 𝑥 is equal to 𝑥 squared minus four 𝑥 plus two. The function is translated 𝑎 units in the 𝑦-direction to create function 𝑔 of 𝑥 is equal to 𝑥 squared minus four 𝑥 plus nine. Find the value of 𝑎.

Well, the question told us that the function is translated 𝑎, so positive 𝑎, units in the 𝑦-direction. Well, that means — and it’s creating function 𝑔 of 𝑥. So, that means 𝑔 of 𝑥 is 𝑓 of 𝑥 plus 𝑎. Now, we were told that 𝑔 of 𝑥 is equal to 𝑥 squared minus four 𝑥 plus nine. So, we can replace that in our equation. And we were told that 𝑓 of 𝑥 is equal to 𝑥 squared minus four 𝑥 plus two. So, we can replace that in our equation. And then, lastly, we’ve just got to add the 𝑎 to the end. So, now, we can rearrange and solve this equation.

Subtracting 𝑥 squared from both sides gives me this. Then, adding four 𝑥 to both sides gives me this. And finally, subtracting two from both sides gives me this. 𝑎 is equal to seven. And if I’ve got enough time, like at the end of an exam, I can check that answer. Translating a function by 𝑎 units in the 𝑦-direction is like doing 𝑓 of 𝑥 plus 𝑎. So, if we reckon the answer 𝑎 is equal to seven, we can work out 𝑓 of 𝑥 plus seven. So, we got 𝑓 of 𝑥, our original 𝑓 of 𝑥 function. And we just add seven to the 𝑦-coordinates like we do there. And that gives us 𝑥 squared minus four 𝑥 plus nine, which indeed is the same as 𝑔 of 𝑥. So, we know we’ve got the right answer.

Our next example then.

𝑓 of 𝑥 is equal to three 𝑥 plus nine. This function is translated positive two units in the 𝑦-direction and 𝑐, or positive 𝑐, units in the 𝑥-direction to form function 𝑔 of 𝑥 is equal to three 𝑥 plus two. Find the value of 𝑐.

Now, to translate a function two units in the 𝑦-direction, we’re gonna map 𝑓 of 𝑥 onto 𝑓 of 𝑥 plus two. We’re adding two to all of the 𝑦-coordinates. And to translate positive 𝑐 units in the 𝑥-direction, we would map our function like this. 𝑓 of 𝑥 is gonna map to 𝑓 of 𝑥 minus 𝑐. So, combining those two different transformations, 𝑓 of 𝑥 is gonna map onto 𝑓 of 𝑥 minus 𝑐 plus two, like this.

Now, we’re told in the question that this forms function 𝑔 of 𝑥. So, this is equal to 𝑔 of 𝑥. So, although they’ve given us an expression for 𝑔 of 𝑥 here, we’re gonna work out an alternative expression. And then, the difference between the two will tell us the value of 𝑐. Now, we know that 𝑓 of 𝑥 is equal to three 𝑥 plus nine. So, we’ve got to try and work out what 𝑓 of 𝑥 minus 𝑐 is. Well, we’re gonna have to replace the 𝑥 in our 𝑓 of 𝑥 function with 𝑥 minus 𝑐. So, instead of three times 𝑥, we’re gonna have three times 𝑥 minus 𝑐. And we’re still gonna have the plus nine on the end of it.

But then, remember, we’ve got to add two. So, simplifying that, 𝑔 of 𝑥 is equal to three 𝑥 minus three 𝑐 plus eleven. But remember, we know that 𝑔 of 𝑥 is equal to three 𝑥 plus two, so we can equate those two things. So, three 𝑥 plus two is equal to three 𝑥 minus three 𝑐 plus eleven. Well, subtracting three 𝑥 from both sides gives me two is equal to negative three 𝑐 plus eleven. Then, I think I’d add three 𝑐 to both sides and then subtract two. And then, I could divide both sides by three, giving me 𝑐 is equal to three.

The function 𝑓 of 𝑥 equals 𝑥 minus five times 𝑥 minus two times 𝑥 plus seven is translated positive five units in the direction of the positive 𝑥-axis. Find an equation for the transformed function.

Well, to translate positive five units in the 𝑥-direction, we need to map 𝑓 of 𝑥 onto 𝑓 of 𝑥 minus five. So, in our function up here 𝑓 of 𝑥, we need to replace 𝑥 with 𝑥 minus five. So, 𝑓 of 𝑥 minus five is equal to well, instead of 𝑥 in our first parentheses there, 𝑥 minus five, we’re gonna use 𝑥 minus five. So, that’s 𝑥 minus five minus five. And again, in the next parentheses, we’re replacing 𝑥 with 𝑥 minus five. So, that becomes 𝑥 minus five minus two. And in the last parentheses again, we’re gonna replace 𝑥 with 𝑥 minus five. So, now, all we have to do is tidy up those parentheses.

Well, the first one, 𝑥 minus five minus another five is 𝑥 minus 10. And in the second one, 𝑥 minus five minus another two is 𝑥 minus seven. And in the last one, 𝑥 minus five plus seven is 𝑥 plus two. So, in fact, that’s our answer. The question only said find an equation. It didn’t tell us to multiply out the parentheses and simplify it down. It just said find an equation. So, technically, we’d have got away with this line here. But I think that’s a little bit cheeky. I think to tidy up a little bit to this answer is probably preferable.

Lastly then, the function 𝑔 of 𝑥 equals 𝑥 minus three times three 𝑥 plus two times four minus 𝑥 is translated two units in the direction of the negative 𝑥-axis. Find an equation for the transformed function.

Now, to translate two units in the direction of the negative 𝑥-axis, that’s the same as translating negative two units in the 𝑥-direction. And to achieve that, we need to map 𝑔 of 𝑥 onto 𝑔 of 𝑥 plus two. And to calculate 𝑔 of 𝑥 plus two, we’re gonna replace 𝑥 with 𝑥 plus two in each case in the function. So, the first parenthesis, instead of 𝑥 minus three, becomes 𝑥 plus two minus three. The second lot becomes, instead of three 𝑥 plus two, it’s three times 𝑥 plus two plus two. And the last parenthesis, instead of four minus 𝑥, is four minus the whole of 𝑥 plus two.

So, we need to be a little bit careful about how we evaluate these. The first one is probably the simplest. 𝑥 plus two minus three is just 𝑥 minus one. Now, the second one, I’m just going to multiply out the parentheses. So, distribute that three across the 𝑥 plus two before I evaluate this. So, that makes it three 𝑥 plus six plus two. And the last parentheses here, I’m taking away 𝑥, but I’m also taking away two. So, I’m just gonna write that out in full. So, that becomes four minus 𝑥 minus two.

So, now, simplifying those two last sets of parentheses. Well, three 𝑥 plus six plus two is three 𝑥 plus eight. And in the last parentheses, I’ve got four take away two gives me two. And then, I’ve got negative 𝑥. So, that’s two take away 𝑥. So, my answer is 𝑔 of 𝑥 plus two is equal to 𝑥 minus one times three 𝑥 plus eight times two minus 𝑥.