### Video Transcript

Which of the following graphs represents the equation π¦ is equal to π₯ squared plus three?

So in order to enable us to solve this problem, what Iβve done is that Iβve actually drawn a sketch of the graph π¦ is equal to π₯ squared. As we can see with π¦ is equal to π₯ squared, we actually have a repeating root. So it just touches the π₯-axis. And it does that at the origin, so the point zero, zero. And thatβs because if π¦ is equal to zero, then π₯ will also be equal to zero. We can also note that the π¦ equals π₯ squared graph is a U-shaped parabola. Okay, brilliant. So we know the basic shape of π¦ equals π₯ squared. Letβs look at ours which is π¦ is equal to π₯ squared plus three.

So our graph is going to be a U-shaped parabola, like π¦ is equal to π₯ squared. But because itβs π¦ is equal to π₯ squared plus three, we know that transformation is taking place. And this transformation is a shift. And weβll have a look at some shift rules to work out which shift it is.

Our first shift rule is that π π₯ plus π, and the plus π is within the parenthesis, is equal to a shift of negative π units in the π₯-axis. What this actually means is that all the π₯-coordinates will actually have π subtracted from them. And the other shift rule is that π π₯ plus π, this time noting that the plus π is actually outside the parenthesis, is equal to a shift of π units in the π¦-axis. And what this means in practice is actually we add π units to all of our π¦-coordinates.

So if we take that back at our function, weβve got π¦ is equal to π₯ squared plus three. Well, this is actually just like the second shift rule that weβve looked at because itβs the same as that π₯ squared. And then weβve added three to it. So therefore, our transformation is going to be a shift of plus three units in the π¦-axis. So what this means in practice is that weβre actually going to add three to the π¦-coordinates of the minimum point cause this is gonna help us identify the graph. Well, our minimum point in a graph of π¦ equals π₯ squared was zero, zero. So therefore, our minimum point is gonna be equal to zero, three because weβve added three units onto our π¦-coordinate.

Okay, great. So with this in mind, letβs choose which graph is the correct graph of this equation. So therefore, we can say that the graph π is the correct graph that represents the equation π¦ is equal to π₯ squared plus three. And we can see that because the minimum point, as we highlighted here, is zero, three. And thatβs what weβre looking for because weβve actually shifted the π₯ squared graph three units in the π¦-axis.