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

𝑓(π‘₯) = π‘₯Β²βˆ’ 4π‘₯ + 2. The function is translated π‘Ž units in the 𝑦-direction to create function 𝑔(π‘₯) = π‘₯Β² βˆ’ 4π‘₯ + 9. Find the value of π‘Ž.

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Video Transcript

𝑓 of π‘₯ is equal to π‘₯ squared minus four π‘₯ plus two. The function is translated π‘Ž units in the 𝑦-direction to create function 𝑔 of π‘₯ is equal to π‘₯ squared minus four π‘₯ plus nine. Find the value of π‘Ž.

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

Subtracting π‘₯ squared from both sides gives me this. Then, adding four π‘₯ to both sides gives me this. And finally, subtracting two from both sides gives me this. π‘Ž 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 π‘Ž units in the 𝑦-direction is like doing 𝑓 of π‘₯ plus π‘Ž. So, if we reckon the answer π‘Ž is equal to seven, we can work out 𝑓 of π‘₯ plus seven. So, we got 𝑓 of π‘₯, our original 𝑓 of π‘₯ function. And we just add seven to the 𝑦-coordinates like we do there. And that gives us π‘₯ squared minus four π‘₯ plus nine, which indeed is the same as 𝑔 of π‘₯. So, we know we’ve got the right answer.

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