Question Video: Identifying Transformations of Cubic Functions in 𝙝-𝙠 Notation Mathematics

Which equation matches the graph? [A] 𝑦 = (π‘₯ βˆ’ 2)Β³ βˆ’ 1 [B] 𝑦 = (π‘₯ + 2)Β³ βˆ’ 1 [C] 𝑦 = (π‘₯ + 2)Β³ + 1 [D] 𝑦 = (π‘₯ βˆ’ 2)Β³ + 1.

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Video Transcript

Which equation matches the graph? Option (A) 𝑦 equals π‘₯ minus two cubed minus one. Option (B) 𝑦 equals π‘₯ plus two cubed minus one. Option (C) 𝑦 equals π‘₯ plus two cubed plus one. Or option (D) 𝑦 equals π‘₯ minus two cubed plus one.

We might begin by noticing that this function looks very similar to the standard cubic function 𝑓 of π‘₯ is equal to π‘₯ cubed, sometimes known as 𝑦 equals π‘₯ cubed. We can sketch 𝑦 equals π‘₯ cubed alongside the given function. The graph of 𝑦 equals π‘₯ cubed has an inflection point at zero, zero. The inflection point of the given function is at negative two, negative one. We could therefore say that the function 𝑦 equals π‘₯ cubed must have been translated two units left and one unit down. Both of these functions have the same steepness, and they have not been reflected, so there are no further transformations.

We can recall that a cubic function in the form 𝑦 is equal to π‘Ž times π‘₯ minus β„Ž cubed plus π‘˜ is a transformation of 𝑦 equals π‘₯ cubed for π‘Ž, β„Ž, and π‘˜ in the real numbers and π‘Ž not equal to zero. In this form, the value of π‘Ž indicates the dilation scale factor and a reflection if π‘Ž is less than zero; there’s a horizontal translation of β„Ž units right and a vertical translation of π‘˜ units up. We perform these transformations with the vertical stretch first, horizontal translation second, and vertical translation third.

In this question, the graph has not been reflected or dilated, so π‘Ž is equal to one. Next, we identified that this graph has a translation of two units left. Because this cubic function form has a horizontal translation of β„Ž units to the right, then this means that our value of β„Ž must be negative. And so β„Ž is equal to negative two. Finally, we identified that there must be a vertical translation of one unit down. Since this form gives us a vertical translation in terms of units upwards, then our value of π‘˜ must be negative. So it’s negative one.

Now all we need to do is fill in the values of π‘Ž, β„Ž, and π‘˜ into this cubic function form. When we simplify, we get the equation 𝑦 equals π‘₯ plus two cubed minus one. This is the equation given in answer option (B).

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