### Video Transcript

The function 𝑓 of 𝑥 is equal to 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.

To do this, we can start by recalling that to translate the graph of a function
positive 𝑎 units in the positive 𝑥-direction, we need to map 𝑓 of 𝑥 onto 𝑓 of
𝑥 minus 𝑎. Since we want to translate positive five units in the direction of the positive
𝑥-axis, we set our value of 𝑎 equal to five and so we want to map 𝑓 of 𝑥 onto 𝑓
of 𝑥 minus five. We can do this by replacing every instance of 𝑥 in the given function with 𝑥 minus
five. We need to replace 𝑥 with 𝑥 minus five everywhere in the given expression. In the first factor, we get 𝑥 minus five minus five. In the second factor, we obtain 𝑥 minus five minus two. And in the third factor, we get 𝑥 minus five plus seven. This is our function 𝑓 of 𝑥 minus five.

We can then simplify each factor in turn. In the first factor, we have 𝑥 minus five minus five is 𝑥 minus 10. In the second factor, we have 𝑥 minus five minus two is 𝑥 minus seven. And in the final factor, we have 𝑥 minus five plus seven is equal to 𝑥 plus
two. This is equal to 𝑓 of 𝑥 minus five. And we know that if we were to sketch both 𝑦 equals 𝑓 of 𝑥 and 𝑦 equals 𝑓 of 𝑥
minus five, we would see that the graph of this function is a translation of the
graph of the function 𝑦 equals 𝑓 of 𝑥 five units in the positive
𝑥-direction.

Hence, our answer is 𝑓 of 𝑥 minus five is equal to 𝑥 minus 10 times 𝑥 minus seven
times 𝑥 plus two.