Video Transcript
The function 𝑦 equals 𝑥 minus one
times two 𝑥 minus three times four minus 𝑥 is translated two units in the
direction of the positive 𝑥-axis. What is the equation of the
resulting function?
In this question, we’re given this
function and the translation. But don’t worry; we don’t need to
actually graph this function in order to be able to answer the question. We can instead apply some logic to
this transformation given that we’re told that it’s a translation in the direction
of the positive 𝑥-axis. The direction of the positive
𝑥-axis is just another way of saying that this translation occurs to the right.
When a graph has shifted right by
𝑎 units, that means that the input has changed to 𝑥 minus 𝑎 for any real value of
𝑎. So this graph, which has shifted
right by two units, means that the input becomes 𝑥 minus two. To find the equation of the
transformed function, then we take every value of 𝑥 and replace it with 𝑥 minus
two. So that’s going to be relatively
simple. For the first set of parentheses,
we’ll have 𝑥 minus two minus one. For the second set of parentheses,
we’ll need to be a little more careful. The original input of 𝑥 was
multiplied by two. So that means that all of 𝑥 minus
two will also need to be multiplied by two.
For the final expression, because
we subtracted 𝑥 from four, we’ll also need to subtract all of 𝑥 minus two from
four. We can then simplify our
equation. The first set of parentheses will
simplify to 𝑥 minus three. Next, when we distribute two across
the parentheses containing 𝑥 minus two, we’ll have two 𝑥 minus four. And subtracting three from that
will give us two 𝑥 minus seven.
In the final expression,
distributing the negative across the parentheses gives us four minus 𝑥 plus two,
which simplifies to six minus 𝑥. We can then give the answer for the
equation as 𝑦 equals 𝑥 minus three times two 𝑥 minus seven times six minus
𝑥.
