Gwen is drawing a graph of the line 𝑦 equals two 𝑥 plus one, from 𝑥 equals zero to 𝑥 equals eight, using the table of values shown. We then got a table of values which shows that when 𝑥 is zero, 𝑦 is one; when 𝑥 is four, 𝑦 is nine; and when 𝑥 is eight, 𝑦 is 17. This is Gwen’s graph. And the question asks us to write down three different mistakes that Gwen has made.
Now, first, you’ll probably recognise that an equation in this format is the equation of a straight line: 𝑦 equals some number times 𝑥 plus another number. The coefficient of 𝑥 the number that 𝑥 is multiplied by tells us the gradient or the slope of the line. And the number on its own, the constant that we’re adding at the end, tells us the 𝑦-intercept, the 𝑦-coordinate of the point at which it cuts the 𝑦-axis. So Gwen has made a good decision to start off with by doing a table of values with three different points. If they all make a straight line, then she knows that she hasn’t made a mistake.
First, let’s check the values in the table of values then. So when 𝑥 equals zero, 𝑦 is equal to two 𝑥 plus one. That’s two times zero plus one. Well, two times zero is zero plus one is equal to one. So yes, when 𝑥 equals zero, 𝑦 equals one. And when 𝑥 equals four, 𝑦 equals two times four plus one. Well, two times four is eight plus one is nine. So yup! That’s correct. And when 𝑥 equals eight, 𝑦 is equal to two times eight — that’s 16 — plus one is 17. So that’s correct. So the table of values looks good. Okay, let’s have a little look at the graph.
Gwen has labelled the 𝑥- and 𝑦-axes correctly, 𝑥 for the horizontal axis and 𝑦 for the vertical axis. The two axes intersect at zero. And the 𝑥-axis, all the numbers from one to nine, are equally spaced. That looks good. But looking at the 𝑦-axis does a bit of a problem. Generally speaking, the numbers are equally spaced. But the numbers four and five are missing. So we’ve spotted our first mistake. Four and five are missing from the 𝑦-axis.
Now, let’s check that she has correctly plotted the points from our table of values. The 𝑥-coordinates tell us how many steps to take along the 𝑥-axis and the 𝑦-coordinates tell us how many steps to take along the 𝑦-axis. When 𝑥 is zero, 𝑦 is one. And we can express that as a coordinate like this: zero for the 𝑥-coordinate, one for the 𝑦-coordinate. Now that should be plotted here on our graph. But it looks like she’s plotted it here. She’s plotted them the wrong way round. She’s taken zero as the 𝑦-coordinate and one as the 𝑥-coordinate. So mistake number two is that Gwen plotted zero, one as one, zero. She got the incorrect placement on the graph.
Now, let’s just quickly check that she plotted the other points correctly. The second point had an 𝑥-coordinate of four and a 𝑦-coordinate of nine. So that’s this point here. Yup, that’s correct. And the third point has an 𝑥-coordinate of eight and a 𝑦-coordinate of 17. So that needs to be plotted eight, 17. That’s up here. So yup, that one is correct as well.
Now the combination of those first two mistakes — the gap in the 𝑦-axis and the incorrectly plotted first point — means that when she tried to join the points up, she couldn’t do that with a straight line, but had to use a curve. Remember we said that the equation 𝑦 equals two 𝑥 plus one describes a straight line. So in orange, I’ve drawn a straight line going through this point and this point extending down in this direction. And we can see that Gwen’s curve doesn’t quite match that straight line. So mistake number three is that Gwen has drawn a curve. But 𝑦 equals two 𝑥 plus one should be a straight line graph.
So remember when you’re plotting a graph for an equation of the form at 𝑦 equals a number times 𝑥 plus another number, it needs to be a straight line. If you plot your three points and your ruler doesn’t go in an exact straight line between them, then that’s an indication that you’ve gone wrong somewhere. And in this format 𝑦 equals something times 𝑥 plus another number, this other number here tells you where your line should cut through the 𝑦-axis. So the alarm bell should have started ringing for Gwen when her curve didn’t cut the 𝑦-axis at one.