Which of the following graphs represents the equation 𝑦 is equal to 𝑥 squared plus three?
So in order to enable us to solve this problem, what I’ve done is that I’ve actually drawn a sketch of the graph 𝑦 is equal to 𝑥 squared. As we can see with 𝑦 is equal to 𝑥 squared, we actually have a repeating root. So it just touches the 𝑥-axis. And it does that at the origin, so the point zero, zero. And that’s because if 𝑦 is equal to zero, then 𝑥 will also be equal to zero. We can also note that the 𝑦 equals 𝑥 squared graph is a U-shaped parabola. Okay, brilliant. So we know the basic shape of 𝑦 equals 𝑥 squared. Let’s look at ours which is 𝑦 is equal to 𝑥 squared plus three.
So our graph is going to be a U-shaped parabola, like 𝑦 is equal to 𝑥 squared. But because it’s 𝑦 is equal to 𝑥 squared plus three, we know that transformation is taking place. And this transformation is a shift. And we’ll have a look at some shift rules to work out which shift it is.
Our first shift rule is that 𝑓 𝑥 plus 𝑎, and the plus 𝑎 is within the parenthesis, is equal to a shift of negative 𝑎 units in the 𝑥-axis. What this actually means is that all the 𝑥-coordinates will actually have 𝑎 subtracted from them. And the other shift rule is that 𝑓 𝑥 plus 𝑎, this time noting that the plus 𝑎 is actually outside the parenthesis, is equal to a shift of 𝑎 units in the 𝑦-axis. And what this means in practice is actually we add 𝑎 units to all of our 𝑦-coordinates.
So if we take that back at our function, we’ve got 𝑦 is equal to 𝑥 squared plus three. Well, this is actually just like the second shift rule that we’ve looked at because it’s the same as that 𝑥 squared. And then we’ve added three to it. So therefore, our transformation is going to be a shift of plus three units in the 𝑦-axis. So what this means in practice is that we’re actually going to add three to the 𝑦-coordinates of the minimum point cause this is gonna help us identify the graph. Well, our minimum point in a graph of 𝑦 equals 𝑥 squared was zero, zero. So therefore, our minimum point is gonna be equal to zero, three because we’ve added three units onto our 𝑦-coordinate.
Okay, great. So with this in mind, let’s choose which graph is the correct graph of this equation. So therefore, we can say that the graph 𝑎 is the correct graph that represents the equation 𝑦 is equal to 𝑥 squared plus three. And we can see that because the minimum point, as we highlighted here, is zero, three. And that’s what we’re looking for because we’ve actually shifted the 𝑥 squared graph three units in the 𝑦-axis.