Question Video: Remake of video with id 624128481423 | Nagwa Question Video: Remake of video with id 624128481423 | Nagwa

Question Video: Remake of video with id 624128481423 Mathematics • Second Year of Secondary School

𝑓(𝑥) = 𝑥² − 4𝑥 + 2. The function is translated “𝑎” units in the 𝑦-direction to create function 𝑔(𝑥) = 𝑥² − 4𝑥 + 9. Find the value of 𝑎.

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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, which is equal to 𝑥 squared minus four 𝑥 plus nine. Find the value of 𝑎.

To begin, we note that we are told that 𝑔 of 𝑥 is a translation of 𝑓 of 𝑥 in the 𝑦-direction. We can then recall that a translation in the 𝑦-direction is given by adding a constant to the function. In general, for 𝑔 of 𝑥 to be a vertical translation of 𝑓 of 𝑥, we must have that 𝑔 of 𝑥 is equal to 𝑓 of 𝑥 plus 𝑎. Then, 𝑔 of 𝑥 is a translation of 𝑓 of 𝑥 𝑎 units in the 𝑦-direction, where it is worth noting that the sign of 𝑎 tells us the direction of the translation. If 𝑎 is positive, then we translate upwards, and if 𝑎 is negative, then we translate downwards.

We can use this equation to find an equation for 𝑎 by substituting the given expressions for 𝑓 of 𝑥 and 𝑔 of 𝑥 into this equation. Substituting the given expressions into the equation yields 𝑥 squared minus four 𝑥 plus nine is equal to 𝑥 squared minus four 𝑥 plus two plus 𝑎. We now want to solve this equation for 𝑎. We can start by subtracting 𝑥 squared from both sides of the equation. At the same time, we can add four 𝑥 to both sides of the equation. And we can also subtract two from both sides of the equation to isolate 𝑎 on the right-hand side of the equation. This then gives us that the value of 𝑎 is equal to seven.

It is also worth noting that we can check our answer by finding an expression for 𝑓 of 𝑥 plus seven, that is, translating 𝑓 of 𝑥 seven units in the positive 𝑦-direction. We substitute the expression we are given for 𝑓 of 𝑥 to get 𝑥 squared minus four 𝑥 plus two plus seven. We then simplify by calculating that two plus seven is equal to nine, giving us 𝑥 squared minus four 𝑥 plus nine. Finally, we see that this is exactly the same as the expression we are given for 𝑔 of 𝑥, confirming that the value of 𝑎 is equal to seven.

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