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In the equation 𝑦 plus three equals 𝑥 squared minus 𝑐, 𝑐 is a constant less than zero.
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Which of the following graphs in the 𝑥𝑦-plane could represent the solutions to this equation?
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So in this question, we’ve got 𝑦 plus three equals 𝑥 squared minus 𝑐.
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Well, the first thing we wanna take a look at is our 𝑥 squared term.
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And that’s because our 𝑥 squared term is positive.
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And if it’s positive 𝑥 squared, this tells us something about the shape of our graph.
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Well, as the 𝑥 squared term is the highest power of 𝑥 that we have, then this term determines the shape of our graph.
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And as we’ve said, it’s positive 𝑥 squared.
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So we know that the coefficient of 𝑥 squared is positive.
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So therefore, our graph’s gonna take the shape of a U-shaped parabola.
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If however it was negative, then it’ll be an inverted U- on an n-shaped parabola.
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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).
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And that’s because both of these answers show graphs that have an inverted U- on an n-shaped parabola.
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So therefore, the coefficient of 𝑥 squared in each of these graphs must be negative.
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Now we’re left with A) and D).
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How can we tell which one of these will represent the solutions to our equation?
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Well, to help us do this, what we’re gonna do is rearrange our equation slightly.
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So what we’re gonna do is subtract three from each side of the equation to make 𝑦 the subject.
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So when we do that, we get 𝑦 is equal to 𝑥 squared minus 𝑐 minus three.
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Well, how is this gonna help us?
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I’m gonna show you how it can help us.
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And first of all, we’re gonna take a look at this part, the part that says negative 𝑐.
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Well, we know that negative 𝑐 must be positive.
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That’s because we’re told 𝑐 is a constant less than zero.
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So therefore, if you have minus a negative, say you subtract a negative, then it becomes positive.
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So therefore, minus 𝑐 must be positive.
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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.
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And at this point, our 𝑥-value is gonna be equal to zero in both situations.
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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.
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So what I’ve done is substituted in 𝑥 is equal to zero.
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And when I do that, I get 𝑦 is equal to zero minus 𝑐 minus three.
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Well, let’s take a look at the part that says negative 𝑐 minus three.
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Well, we’ve already said that negative 𝑐 must be positive.
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So therefore, it’s gonna be greater than zero.
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So therefore, negative 𝑐 minus three must be greater than negative three.
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That’s because if we had, for instance, zero minus three, then this would give us negative three.
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And we already know that negative 𝑐 must be greater than zero.
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So that means the result of negative 𝑐 minus three must also be greater and this time than negative three.
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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.
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And three is greater than negative three.
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So this would work.
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However, if we look at D), we’ve got negative four.
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And negative four is less than negative three.
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So therefore, this wouldn’t work.
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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).