Consider the function 𝐹 of 𝑥 equals 𝑥 plus three over 𝑥 minus one. The function 𝑦 equals 𝐹 of 𝑥 is graphed on the 𝑥𝑦-plane shown. If the equation 𝑦 equals 𝑥 plus three is drawn on the same set of axes, which of the following would be a point of intersection of the two graphs? The choices are A) two, five; B) one, zero; C) two, zero; or D) zero, two.
So to work this out, what we’re going to do is plot the equation 𝑦 equals 𝑥 plus three on the same axes. So in order to help us do that, what I’ve done is drawn a table. So I’ve got my 𝑥-values and my 𝑦-values. And now I’m gonna choose three 𝑥-values. And the 𝑥-values I’ve chosen are negative 15, zero, and 15. And I’ve chosen these because they’re easy to make out on our scale that we have. You could choose any three points you wanted to on the line to help you draw the straight line. In fact, you only need two points. I’ve just chosen three just to make sure.
So now what I need to do is work out what the corresponding 𝑦-values are. And all I need to do to find them out is to add three. And that’s because our equation is 𝑦 equals 𝑥 plus three. And when I do that, the values I get are negative 12, three, and 18. And remembering that I get negative 12 for the first value. And that’s because if we’ve got negative 15 and we add three, it means that we’d be moving right up the number line. So we get less negative. So that’s why we get to negative 12.
So now what I do is I plot my points. So I’ve got my first point: negative 15, negative 12. So the next point is zero, three that I’ve also plotted on the graph. And that’s zero on the 𝑥-axis and three on the 𝑦-axis. And then, finally, I have 15, 18. So I’ve plotted that. So then all I need to do is draw the straight line and label my line, which is 𝑦 equals 𝑥 plus three.
And if we take a look at our graphs, we can see that there are two points of intersection. So that means that there are two points where the graph of the function 𝐹 of 𝑥 equals 𝑥 plus three over 𝑥 minus one meets the graph of the equation 𝑦 equals 𝑥 plus three. So now if we take a look at what the coordinates of our points are, we can see that the right-hand point of intersection is gonna have two along the 𝑥-axis and five in the 𝑦-axis. So therefore, the coordinates are going to be two, five. And then the other point on the left-hand side is going to be negative three in the 𝑥-axis and zero in the 𝑦-axis. So it’s negative three, zero.
And what we can do to check that I’ve read off the values correctly is substitute the values into 𝑦 equals 𝑥 plus three to make sure that they fit. So if I take the right-hand point, I’ve got 𝑥 is equal to two and 𝑦 is equal to five. Well, if I substitute those in, I get five is equal to two plus three. And this is correct. So therefore, I’ve read that point off correctly.
So for the next point, we’ve got negative three, zero. So 𝑥 is negative three. 𝑦 is zero. Substitute this in, we get zero is equal to negative three plus three, which is also correct. So I’ve also read this point off correctly.
Well, we can double-check by trying our values in the other function. So if we do that, we’ve got 𝐹 of 𝑥 is equal to 𝑥 plus three over 𝑥 minus one. So if we substitute in two five — so 𝑥 is equal to two, 𝑦 is equal to five — we get five is equal to two plus three over two minus one, which will give us five over one. And that’s cause two plus three is five and two minus one is one. So that would give us five, which is what we’ve been looking for. So this is correct.
So then if we try negative three, zero as well, we get zero is equal to negative three plus three over negative three minus one. And this gives us zero over negative four. Well, we can actually see that zero divided by anything is always zero. So this is also correct. So we’ve definitely read the two points off correctly.
So therefore, we can come back to the question. And we can say that the correct coordinate for a point of intersection from the options that we’ve been given is A. And that’s two, five. And this is correct because this is one of our points of intersection that we’ve shown.
It’s also worth mentioning with this question, even though we found the final answer and we’ve shown the method how, there is actually a shortcut. And that’s because if we look at the equation, we have 𝑦 is equal to 𝑥 plus three. So therefore, the 𝑦-coordinate must be three bigger or three greater than the 𝑥-coordinate. And that’s because we’re looking for a point of intersection, so a point that lies on this equation line as well.
And if we take a look at our four points, well, for point A, the correct point, we can see that the 𝑦-coordinate is three greater than the 𝑥-coordinate. If we look at point B, the 𝑦-coordinate is actually less than the 𝑥-coordinate. Look at point C. Again, the 𝑦-coordinate is less than the 𝑥-coordinate. And if we look at point D, the 𝑦-coordinate is two greater than the 𝑥-coordinate. So we can say that 𝑦 equals 𝑥 plus two. So that’s just a quick shortcut that could’ve saved you time on this particular question.