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

The function π¦ = π(π₯) is translated eight down. Write, in terms of π(π₯), the equation of the translated graph.

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### Video Transcript

The function π¦ is equal to π of π₯ is translated eight down. Write, in terms of π of π₯, the equation of the translated graph.

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

Now, remember, in our graphs the π¦-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 π¦-coordinates. So, we want the outputs of our function to change. Letβs see what would happen if we want to add π to our function. Now, we can see for the same value of π₯, we get π of π₯. However, we also add a value of π, our outputs of π larger. So, now, we can see weβve increased the π¦-coordinate of every single point on our curve up by π. This is a translation of π units upwards.

And we can also ask the question, βwhat would happen if we were to subtract π instead?β This time, for the same input value of π₯, we can now see our π¦-coordinate is π lower. Weβve translated our curve π units downwards.

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

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