Video Transcript
Which of the following could be the equation of the given graph? Is it A) 𝑦 equals 𝑥 minus one multiplied by 𝑥 plus three multiplied by 𝑥 minus four? Is it B) 𝑦 equals 𝑥 plus one all squared multiplied by 𝑥 minus three multiplied by 𝑥 plus four? Option C) 𝑦 equals 𝑥 plus one multiplied by 𝑥 minus three multiplied by 𝑥 plus four? Or option D) 𝑦 equals 𝑥 minus one all squared multiplied by 𝑥 plus three multiplied by 𝑥 minus four?
There are two ways of approaching this question. We could look at the four options and work out what the roots, or solutions, are for each equation. Alternatively, we could look at the graph first and look at the points of intersection on the 𝑥- and 𝑦-axis. The graph crosses or touches the 𝑥-axis at negative three, positive one, and positive four. This means that we have solutions, or roots, at negative three, one, and four. As the graph does not cross the 𝑥-axis at one, it just touches it. This is actually a double root, or repeated root.
We know that if 𝑥 equals 𝑎 is a root, or solution, then 𝑥 minus 𝑎 is a factor. As 𝑥 equals negative three is a root, 𝑥 plus three is a factor. One of our brackets must be 𝑥 plus three. In the same way, as 𝑥 equals four is a root, 𝑥 minus four must be a factor. One of the other brackets must be 𝑥 minus four. As 𝑥 equals one was a double root, then a factor must be 𝑥 minus one squared. The double, or repeated, root indicates that the bracket must appear twice.
Our solution must contain the brackets 𝑥 plus three, 𝑥 minus one squared, and 𝑥 minus four. The correct answer is, therefore, option D, 𝑦 is equal to 𝑥 minus one squared multiplied by 𝑥 plus three multiplied by 𝑥 minus four. We can check this answer by considering the equation first. When the graph crosses or touches the 𝑥 axis, we know that the 𝑦-coordinate is nought. Therefore, we can write zero is equal to 𝑥 minus one squared multiplied by 𝑥 plus three multiplied by 𝑥 minus four. This gives us three possible solutions, 𝑥 minus one all squared is equal to zero, 𝑥 plus three is equal to zero, or 𝑥 minus four is equal to zero.
Square rooting both sides of the first equation and then adding one to both sides gives us an answer of 𝑥 equals one. Subtracting three from both sides of the second equation gives us 𝑥 equals negative three. Finally, adding four to the third equation gives us 𝑥 is equal to four. This confirms our three values for 𝑥 negative three, one, and four with a double or repeated root at 𝑥 equals one.
Option A would also give us three roots of one, negative three, and four. However, we do not have the double root at 𝑥 equals one. Option B gives us a double root at 𝑥 equals negative one and a single root at 𝑥 equals three and 𝑥 equals negative four. These are not the points of intersection on the graph. Option C also gives us roots at negative one, three, and negative four. This, once again, does not match the graph.
We can also consider the points of intersection on the 𝑦-axis. This occurs when 𝑥 is equal to zero. Substituting 𝑥 equals zero into our equation gives us 𝑦 is equal to zero minus one all squared multiplied by zero plus three multiplied by zero minus four. Zero minus one is equal to negative one. And squaring this gives us positive one, as multiplying a negative by a negative gives us a positive answer. Zero plus three is equal to three.
And zero minus four is equal to negative four. We need to multiply one, three, and negative four. One multiplied by three is equal to three. And multiplying this by negative four gives us negative 12. This means that the graph must cross the 𝑦-axis at the coordinate zero, negative 12. The 𝑦-intercept is negative 12. We can see this is correct as the graph crosses the 𝑦-axis two squares below negative 10 and three squares above negative 15. We have, therefore, proved that the correct equation for the given graph is 𝑦 is equal to 𝑥 minus one all squared multiplied by 𝑥 plus three multiplied by 𝑥 minus four.
