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Which of the following graphs represents the equation 𝑦 equals 𝑥 plus four multiplied by 𝑥 minus two?
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Well, the first thing we want to do is we want to set our equation equal to zero.
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So, we got 𝑥 plus four multiplied by 𝑥 minus two is equal to zero.
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And we want to do this because when 𝑦 is equal to zero, this will tell us where it crosses the 𝑥-axis.
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Well, we’ll find the solutions, zeroes, or roots of our equation.
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So, for this equation to be equal to zero, one of our parentheses is gonna be equal to zero.
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So, either 𝑥 plus four is equal to zero, which would mean that 𝑥 would be equal to negative four, or 𝑥 minus two is gonna be equal to zero.
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So therefore, 𝑥 will be equal to two or positive two.
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Okay, great.
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So, we now know the roots or solutions to our equation, and they are 𝑥 is equal to negative four or two.
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And that’s when 𝑦 is equal to zero.
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Okay, great.
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So, can we rule out or in any of our graphs using this information?
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Well, it could be graph A because we can see that our curve crosses the 𝑥-axis at negative four and two.
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Well, it cannot be graph B.
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And that’s because we can see graph B, the curve crosses the 𝑥-axis at negative two, four.
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So, the sign is the opposite way around.
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Again, we can rule out C.
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And this time is because we’ve got the 𝑥-axis being crossed at negative five and a half and negative two and a half.
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Well, graph D, the curve crosses it at negative four and two.
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So, graph D could be the correct graph.
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Again, we can rule out graph E.
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And we could rule out graph E because graph E doesn’t, in fact, cross the 𝑥-axis at all.
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So therefore, it would give us no solutions to the equation if it was equal to zero.
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So, now we need to decide how we’re gonna distinguish between graph A and graph D.
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Well, if we would have a look at the equation of our curve again, we’ve got 𝑦 is equal to 𝑥 plus four multiplied by 𝑥 minus two.
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Now, if we were to distribute across our parentheses, well our first step would be to multiply the two 𝑥s together.
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And what this would give us is 𝑥 squared.
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So, we don’t even have to carry on with distributing across our parentheses because this is the only bit of information we need.
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Because, as I said, the first term is going to be 𝑥 squared in our quadratic.
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Well, if we have a positive 𝑥 squared term so the coefficient of 𝑥 squared is positive, then we know that our parabola is gonna be U-shaped.
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However, if we have a negative 𝑥 squared term so the coefficient is negative, then it’s gonna be an inverted U- or N-shaped parabola.
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Well, we could see that our 𝑥 squared is positive.
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So therefore, we can rule out graph D because this is an inverted U- or N-shaped parabola.
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And therefore, the correct graph must be the graph which is graph A because we have a U-shaped parabola, which is because we’ve got a positive 𝑥 squared term.
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And we’ve shown that it crosses the 𝑥-axis at negative four and two.
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But what I can do is a quick check.
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And the check I’m going to do is I’m gonna work out what the 𝑦-intercept is going to be.
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Well, in order for me to find out what the 𝑦-intercept is going to be, what I’m gonna do is substitute in 𝑥 is equal to zero.
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Because when 𝑥 is equal to zero, that means we’re gonna be cutting the 𝑦-axis.
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Well therefore, what we’re gonna have is zero plus four multiplied by zero minus two.
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So, we’re gonna have 𝑦 is equal to four multiplied by negative two.
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Well, this will give us 𝑦 is equal to negative eight.
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So therefore, we should have a 𝑦-intercept at negative eight.
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And if we look back at the graph, this is the case.
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So, we can definitely confirm that graph A is the correct graph.