### Video Transcript

In the equation 𝑦 plus three equals 𝑥 squared minus 𝑐, 𝑐 is a constant less than zero. Which of the following graphs in the 𝑥𝑦-plane could represent the solutions to this equation?

So in this question, we’ve got 𝑦 plus three equals 𝑥 squared minus 𝑐. Well, the first thing we wanna take a look at is our 𝑥 squared term. And that’s because our 𝑥 squared term is positive. And if it’s positive 𝑥 squared, this tells us something about the shape of our graph.

Well, as the 𝑥 squared term is the highest power of 𝑥 that we have, then this term determines the shape of our graph. And as we’ve said, it’s positive 𝑥 squared. So we know that the coefficient of 𝑥 squared is positive. So therefore, our graph’s gonna take the shape of a U-shaped parabola. If however it was negative, then it’ll be an inverted U- on an n-shaped parabola.

Well, because we know that it’s going to be a U-shaped parabola, it means that we can cancel out two other possible answers, answer B) and answer C). And that’s because both of these answers show graphs that have an inverted U- on an n-shaped parabola. So therefore, the coefficient of 𝑥 squared in each of these graphs must be negative.

Now we’re left with A) and D). How can we tell which one of these will represent the solutions to our equation? Well, to help us do this, what we’re gonna do is rearrange our equation slightly. So what we’re gonna do is subtract three from each side of the equation to make 𝑦 the subject. So when we do that, we get 𝑦 is equal to 𝑥 squared minus 𝑐 minus three. Well, how is this gonna help us?

I’m gonna show you how it can help us. And first of all, we’re gonna take a look at this part, the part that says negative 𝑐. Well, we know that negative 𝑐 must be positive. That’s because we’re told 𝑐 is a constant less than zero. So therefore, if you have minus a negative, say you subtract a negative, then it becomes positive. So therefore, minus 𝑐 must be positive.

So now if we take a look at our graphs, we can see that there’s a point on both of these graphs where our curve crosses the 𝑦-axis. And at this point, our 𝑥-value is gonna be equal to zero in both situations. So therefore, to see where the graph we’re looking for will cross the 𝑦-axis, we can see what would happen when 𝑥 is equal to zero. So what I’ve done is substituted in 𝑥 is equal to zero. And when I do that, I get 𝑦 is equal to zero minus 𝑐 minus three.

Well, let’s take a look at the part that says negative 𝑐 minus three. Well, we’ve already said that negative 𝑐 must be positive. So therefore, it’s gonna be greater than zero. So therefore, negative 𝑐 minus three must be greater than negative three. That’s because if we had, for instance, zero minus three, then this would give us negative three. And we already know that negative 𝑐 must be greater than zero. So that means the result of negative 𝑐 minus three must also be greater and this time than negative three.

So if we take a look at what we’ve got, for our 𝑦-intercept, we look at A), so the value of 𝑦 when 𝑥 equals zero is three. And three is greater than negative three. So this would work. However, if we look at D), we’ve got negative four. And negative four is less than negative three. So therefore, this wouldn’t work.

So therefore, if we’ve got the equation 𝑦 plus three equals 𝑥 squared minus 𝑐 and 𝑐 is a constant less than zero, the graph in the 𝑥𝑦-plane that could represent the solutions to the equation is graph A).