Question Video: Sketching Curves from their Roots | Nagwa Question Video: Sketching Curves from their Roots | Nagwa

Question Video: Sketching Curves from their Roots Mathematics • Second Year of Secondary School

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Which of the following is the graph of π(π₯) = β(π₯ β 2)Β³? [A] Graph A [B] Graph B [C] Graph C [D] Graph D [E] Graph E

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Video Transcript

Which of the following is the graph of π of π₯ is equal to negative π₯ minus two cubed?

In this question, weβre given a cubic function. So letβs compare this with the standard cubic function π of π₯ is equal to π₯ cubed. We can draw a quick sketch of this function. Letβs recall that a cubic function in the form π of π₯ is equal to π times π₯ minus β cubed plus π is the transformation of π of π₯ equals π₯ cubed for π, β, and π in the real numbers and π not equal to zero.

Here, π represents a dilation or reflection, β gives the number of units the graph has been translated in the horizontal direction, and π is the number of units the graph is translated in the vertical direction. We perform these transformations with the vertical dilation first, horizontal translation second, and vertical translation third.

So letβs consider the function we were given. Since π of π₯ is equal to negative π₯ minus two cubed, then that means that π is equal to negative one. This indicates that there is no dilation, or rather a dilation of scale factor one. However, since π is negative, this means there is a reflection of the graph in the π₯-axis. If we perform just the reflection, then the graph would look like this in pink.

Next, in the given function, β is equal to two. So this means that there is a translation of two units to the right. This moves the inflection point from zero, zero to two, zero. Therefore, the function would look something like this. Itβs always a good idea to mark on any important information onto a sketch. And of course, we know that this graph crosses the π₯-axis at two, zero.

We can then give the answer that the graph which shows π of π₯ is equal to negative π₯ minus two cubed is that given in option (E).

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