Video: Finding the Equation of a Translated Graph Using Horizontal and Vertical Shifts

The function 𝑦 = 𝑓(π‘₯) is translated eight down. Write, in terms of 𝑓(π‘₯), the equation of the translated graph.

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Video Transcript

The function 𝑦 is equal to 𝑓 of π‘₯ is translated eight down. Write, in terms of 𝑓 of π‘₯, the equation of the translated graph.

In this question, we’re given a function 𝑓 of π‘₯. And we’re given the curve of this function, 𝑦 is equal to 𝑓 of π‘₯. And we’re told that this curve is translated eight down. Remember, this just means it’s translated eight units vertically downwards. We need to find an equation for the translated graph in terms of our original function 𝑓 of π‘₯. To answer this question, we’re first going to need to recall how do we represent translations in terms of our functions. And to do this, we’ll start by considering the curve 𝑦 is equal to 𝑔 of π‘₯.

Now, remember, in our graphs the 𝑦-axis is the vertical axis, and the π‘₯-axis is the horizontal axis. So, if we want to translate a curve upwards or downwards, we need to affect its 𝑦-coordinates. So, we want the outputs of our function to change. Let’s see what would happen if we want to add 𝑏 to our function. Now, we can see for the same value of π‘₯, we get 𝑔 of π‘₯. However, we also add a value of 𝑏, our outputs of 𝑏 larger. So, now, we can see we’ve increased the 𝑦-coordinate of every single point on our curve up by 𝑏. This is a translation of 𝑏 units upwards.

And we can also ask the question, β€œwhat would happen if we were to subtract 𝑏 instead?” This time, for the same input value of π‘₯, we can now see our 𝑦-coordinate is 𝑏 lower. We’ve translated our curve 𝑏 units downwards.

And this is actually enough information to help us answer this question. We want to translate the curve 𝑦 is equal to 𝑓 of π‘₯ eight units downwards, and we can see this is the second case. We want to reduce all of our 𝑦-coordinates by our value of eight. So, we need to set our value of 𝑏 equal to eight in the second example. So, by setting 𝑏 equal to eight and using 𝑓 of π‘₯ instead of 𝑔 of π‘₯ in our second example, we get 𝑦 is equal to 𝑓 of π‘₯ minus eight, which is our final answer.

Therefore, we were able to show if the function 𝑦 is equal to 𝑓 of π‘₯ is translated eight down, then we can write the equation of the translated graph in terms of 𝑓 of π‘₯. We get 𝑦 is equal to 𝑓 of π‘₯ minus eight.

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