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