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.
