Video: Finding the Number of Units That a Function Is Translated in the 𝑦-Direction to Another One

Video Transcript

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.