Question Video: Figuring Out the Rule of a Quadratic Function given Its Graph Mathematics

Video Transcript

Which of the following graphs represents 𝑓 𝑥 is equal to 𝑥 minus one all squared?

To help us to understand this question better, I’ve drawn a little sketch. And this sketch shows that the function 𝑓 𝑥 is equal to 𝑥 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 that’s the shape of the curve we’d get if it was 𝑥 squared.

However, our function this time is 𝑥 minus one squared. And to help us with this, what we’re gonna have look at is a little couple of rules for translation, so when we’re translating a graph. Our first translation rule is that if we had 𝑓 𝑥 plus 𝑎, it’s equal to a shift of 𝑎 units in the 𝑦-direction.

And what it is important to note here is that the plus 𝑎 is actually outside the parentheses. So it’s- sort of not in the parentheses with the 𝑥. So we know this is in the 𝑦-direction. Our second rule is that 𝑓 𝑥 plus 𝑎 is equal to a shift of negative 𝑎 units in the 𝑥-direction.

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

Great! So now we know this. We can have a look at the function that we’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 𝑥 minus one in the parentheses all squared is gonna link to our second rule.

So therefore, this means that we’re gonna get a shift of plus one unit in the 𝑥-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 𝑥 squared graph. What’s gonna happen to this?

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

So I’ve sketched what happens on our original graph. So if you have a look there, we can see that actually our 𝑥-coordinates have all increased by one. So it’s been a shift to the right. So now which one of our graphs will that apply to?

Well we can see that it’s graph 𝑏 because that actually has the point where it touches the 𝑥-axis is actually at one because it shifted one to the right. So therefore, we can say that graph 𝑏 represents 𝑓 𝑥 is equal to 𝑥 minus one all squared.