Question Video: Determining the Function of a Translated Graph Mathematics

Video Transcript

The graph of the function 𝑓, 𝑦 is equal to 𝑓 of 𝑥 is translated four units in the positive 𝑦-direction. Write, in terms of 𝑓 of 𝑥, the equation of the translated graph.

We’re told that the graph 𝑦 is equal to 𝑓 of 𝑥 of the function 𝑓 is translated four units in the positive 𝑦-direction. So let’s see how this looks with respect to a particular graph. Suppose, for example, our function looks like the one shown. Now, if this function is translated four units in the positive 𝑦-direction, this means that the whole function is moved up by four units. But what does this mean with respect to our function 𝑦 is equal to 𝑓 of 𝑥?

Well, to see what’s happening, we can look at some specific points. And for convenience, we choose points where both the input and output are whole numbers. So, for example, when 𝑥 is negative two, 𝑦 equals 𝑓 of 𝑥 is positive two. When 𝑥 is negative one, 𝑦 is negative one. When 𝑥 is zero, 𝑦 is negative two. And when 𝑥 is one, 𝑦 is negative one. And 𝑥 is two; 𝑦 is also two. But now what about our new function, the translated function, which we’ll call 𝑔 of 𝑥?

When 𝑥 is negative two in this case, 𝑔 of 𝑥 is positive six. When 𝑥 is negative one, 𝑔 of 𝑥 is positive three. When 𝑥 is zero, 𝑔 of 𝑥 is positive two. When 𝑥 is one, 𝑔 of 𝑥 is three. And when 𝑥 is two, 𝑔 of 𝑥 is six. So let’s compare these values with our outputs for 𝑓 of 𝑥. So, when 𝑥 is negative two, we see that 𝑔 of 𝑥 is six, which is two plus four, and that’s equal to 𝑓 of negative two plus four. And similarly, when 𝑥 is negative one, we see that 𝑔 of negative one is 𝑓 of negative one plus four. That’s minus one plus four. When 𝑥 is zero, 𝑔 of zero is two, which is equal to negative two plus four, and that’s 𝑓 of zero plus four. And when 𝑥 is one, 𝑔 of one is three, which is negative one plus four, and that’s 𝑓 of one plus four. And finally, when 𝑥 is equal to two, 𝑔 of two is equal to two plus four. That’s 𝑓 of two plus four, which is six.

So we see that in every case, 𝑔 of 𝑥 is equal to the value of 𝑓 of 𝑥 plus four. That is, for all of our 𝑥-values, the outcome after the translation by four units in the positive 𝑦-direction is 𝑦 is equal to 𝑓 of 𝑥 plus four. And this is the equation of the translated graph.