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Question Video: Finding the Transformed Function in the Direction of the Positive π‘₯-Axis Mathematics • 10th Grade

The function 𝑓(π‘₯) = (π‘₯ βˆ’ 5)(π‘₯ βˆ’ 2)(π‘₯ + 7) is translated +5 units in the direction of the positive π‘₯-axis. Find an equation for the transformed function.

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Video Transcript

The function 𝑓 of π‘₯ is equal to π‘₯ minus five times π‘₯ minus two times π‘₯ plus seven is translated positive five units in the direction of the positive π‘₯-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 π‘Ž units in the positive π‘₯-direction, we need to map 𝑓 of π‘₯ onto 𝑓 of π‘₯ minus π‘Ž. Since we want to translate positive five units in the direction of the positive π‘₯-axis, we set our value of π‘Ž equal to five and so we want to map 𝑓 of π‘₯ onto 𝑓 of π‘₯ minus five. We can do this by replacing every instance of π‘₯ in the given function with π‘₯ minus five. We need to replace π‘₯ with π‘₯ minus five everywhere in the given expression. In the first factor, we get π‘₯ minus five minus five. In the second factor, we obtain π‘₯ minus five minus two. And in the third factor, we get π‘₯ minus five plus seven. This is our function 𝑓 of π‘₯ minus five.

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

Hence, our answer is 𝑓 of π‘₯ minus five is equal to π‘₯ minus 10 times π‘₯ minus seven times π‘₯ plus two.

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