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Which of the following graphs represents π π₯ is equal to π₯ minus one all squared?
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To help us to understand this question better, Iβve drawn a little sketch.
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And this sketch shows that the function π π₯ is equal to π₯ squared.
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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.
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So thatβs the shape of the curve weβd get if it was π₯ squared.
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However, our function this time is π₯ minus one squared.
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And to help us with this, what weβre gonna have look at is a little couple of rules for translation, so when weβre translating a graph.
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Our first translation rule is that if we had π π₯ plus π, itβs equal to a shift of π units in the π¦-direction.
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And what it is important to note here is that the plus π is actually outside the parentheses.
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So itβs- sort of not in the parentheses with the π₯.
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So we know this is in the π¦-direction.
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Our second rule is that π π₯ plus π is equal to a shift of negative π units in the π₯-direction.
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And the two key points to notice this time are firstly that the plus π is within this- side the parentheses and secondly is a shift of negative π units in the π₯-direction.
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So itβs important to know that whenever weβre dealing with π₯-direction translations, that first of all, like we said, the π will be inside the parentheses.
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And second of all, it does the opposite of what you think.
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So itβs actually gonna be a shift of negative π.
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Great!
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So now we know this.
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We can have a look at the function that weβve got and pick which graph would be suitable.
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Well, looking back at our original function, we can say okay well therefore our function of π₯ minus one in the parentheses all squared is gonna link to our second rule.
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So therefore, this means that weβre gonna get a shift of plus one unit in the π₯-direction.
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But what does this mean in actual practice?
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So what does this mean to our graph?
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Cause we look on the top right-hand side, we can see our π₯ squared graph.
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Whatβs gonna happen to this?
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Well, in practice, that actually means that all our π₯-coordinates are gonna be increased by one.
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And the reason that itβs increased by one and add one, cause remember if we look at the function, our π-value is negative one.
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And remembering the rule, it says a shift of negative π.
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So negative negative one gives us plus one, or π₯-coordinates have all increased by one.
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So Iβve sketched what happens on our original graph.
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So if you have a look there, we can see that actually our π₯-coordinates have all increased by one.
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So itβs been a shift to the right.
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So now which one of our graphs will that apply to?
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Well we can see that itβs graph π because that actually has the point where it touches the π₯-axis is actually at one because it shifted one to the right.
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So therefore, we can say that graph π represents π π₯ is equal to π₯ minus one all squared.