Question Video: Figuring Out the Rule of a Quadratic Function given Its Graph Mathematics • 10th Grade

Which of the following graphs represents ๐‘“(๐‘ฅ) = (๐‘ฅ โˆ’ 1)ยฒ? [A] Graph a [B] Graph b [C] Graph c [D] Graph d.

03:14

Video Transcript

Which of the following graphs represents ๐‘“ ๐‘ฅ is equal to ๐‘ฅ minus one all squared?

To help us to understand this question better, Iโ€™ve drawn a little sketch. And this sketch shows that the function ๐‘“ ๐‘ฅ is equal to ๐‘ฅ squared. So as you can see, with this function, what we actually have is a U-shaped parabola, which is symmetrical and actually touches the origin at zero. So thatโ€™s the shape of the curve weโ€™d get if it was ๐‘ฅ squared.

However, our function this time is ๐‘ฅ minus one squared. And to help us with this, what weโ€™re gonna have look at is a little couple of rules for translation, so when weโ€™re translating a graph. Our first translation rule is that if we had ๐‘“ ๐‘ฅ plus ๐‘Ž, itโ€™s equal to a shift of ๐‘Ž units in the ๐‘ฆ-direction.

And what it is important to note here is that the plus ๐‘Ž is actually outside the parentheses. So itโ€™s- sort of not in the parentheses with the ๐‘ฅ. So we know this is in the ๐‘ฆ-direction. Our second rule is that ๐‘“ ๐‘ฅ plus ๐‘Ž is equal to a shift of negative ๐‘Ž units in the ๐‘ฅ-direction.

And the two key points to notice this time are firstly that the plus ๐‘Ž is within this- side the parentheses and secondly is a shift of negative ๐‘Ž units in the ๐‘ฅ-direction. So itโ€™s important to know that whenever weโ€™re dealing with ๐‘ฅ-direction translations, that first of all, like we said, the ๐‘Ž will be inside the parentheses. And second of all, it does the opposite of what you think. So itโ€™s actually gonna be a shift of negative ๐‘Ž.

Great! So now we know this. We can have a look at the function that weโ€™ve got and pick which graph would be suitable. Well, looking back at our original function, we can say okay well therefore our function of ๐‘ฅ minus one in the parentheses all squared is gonna link to our second rule.

So therefore, this means that weโ€™re gonna get a shift of plus one unit in the ๐‘ฅ-direction. But what does this mean in actual practice? So what does this mean to our graph? Cause we look on the top right-hand side, we can see our ๐‘ฅ squared graph. Whatโ€™s gonna happen to this?

Well, in practice, that actually means that all our ๐‘ฅ-coordinates are gonna be increased by one. And the reason that itโ€™s increased by one and add one, cause remember if we look at the function, our ๐‘Ž-value is negative one. And remembering the rule, it says a shift of negative ๐‘Ž. So negative negative one gives us plus one, or ๐‘ฅ-coordinates have all increased by one.

So Iโ€™ve sketched what happens on our original graph. So if you have a look there, we can see that actually our ๐‘ฅ-coordinates have all increased by one. So itโ€™s been a shift to the right. So now which one of our graphs will that apply to?

Well we can see that itโ€™s graph ๐‘ because that actually has the point where it touches the ๐‘ฅ-axis is actually at one because it shifted one to the right. So therefore, we can say that graph ๐‘ represents ๐‘“ ๐‘ฅ is equal to ๐‘ฅ minus one all squared.

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