Point 𝐴 has been plotted on this coordinate grid. Write the coordinates of point 𝐴.
Remember, coordinates are pairs of numbers that describe a position of a point on a set of axes. The first number in the pair represents the value of the 𝑥-coordinate, and the second number represents the value of the 𝑦-coordinate.
We can use the phrase “along the corridor, up the stairs” to help us remember this. The 𝑥-coordinate represents the corridor. We either moved left or right depending on the value of this coordinate. The 𝑦-coordinate represents the stairs. Depending on the value of this coordinate, we move up or down.
For point 𝐴, we first move along the corridor two spaces left. That gives us an 𝑥-coordinate of negative two. To get a positive 𝑥-coordinate, we would need to have moved right. Next, we move three spaces up the stairs. That gives us a 𝑦-coordinate of positive three. Remember, if we’d moved down the stairs, that would have given us a negative 𝑦-coordinate. The coordinates of point 𝐴 are negative two, three.
The point 𝐵 has coordinates seven, negative two. Draw and label this point on the grid.
Once again, we’ll use “along the corridor, up the stairs.” A positive seven for the 𝑥-coordinate tells us to go right along the corridor seven units. The negative two tells us to go down the stairs two units. This takes us to point 𝐵 here.
Part c of this question: which of the two points lies on the straight line with equation 𝑦 is equal to negative two?
Let’s first plot the line 𝑦 is equal to negative two. This is so called because every coordinate on this line has a 𝑦-value of negative two. For example, this point here has an 𝑥-coordinate of two and a 𝑦-coordinate of negative two. And this coordinate over here has an 𝑥-coordinate of negative three, and once again it has a 𝑦-coordinate of negative two. The line 𝑦 equals negative two is a horizontal line that passes through the 𝑦-axis at negative two.
Be careful! A common mistake here is to think that because the line has an equation of 𝑦 equals negative two that it runs parallel to the 𝑦-axis. In fact, any line that runs vertically will be of the form 𝑥 is equal to some value.
Once we’ve drawn this line, we can see that point 𝐵 lies on the line 𝑦 is equal to negative two. Remember how we said earlier that all coordinates on the line 𝑦 equals negative two have a 𝑦-value of negative two. We’ve already seen that the coordinate for 𝐵 is seven, negative two. The 𝑦-value of this coordinate is negative two. So we could have got the answer to this question straight from the coordinate itself.
d) Draw the line 𝑦 equals 𝑥 plus three on the grid.
There are two ways of doing this. The first is to construct a table of values. We need to ensure we choose a minimum of three values for 𝑥. Positive values are better as they’re easier to work with. Let’s choose four values. We can choose zero, one, two, and three.
Once we have these, we substitute them into the equation 𝑦 is equal to 𝑥 plus three. When 𝑥 is zero, our equation for 𝑦 becomes 𝑦 is equal to zero plus three, which is three. When 𝑥 is one, it becomes 𝑦 is equal to one plus three, which is four. When 𝑥 is equal to two, it becomes 𝑦 is equal to two plus three, which is five. And when 𝑥 is equal to three, the equation becomes 𝑦 is equal to three plus three, which is six.
Then we can plot these onto the graph, once again remembering to go along the corridor, up the stairs. The first coordinate lies at zero, three then one, four; two, five; and three, six. Notice how these points are all in one single straight line. That’s a good indicator that we’ve worked out our values correctly. We mustn’t forget to join them with a nice long line that passes from one side of our axes to the other.
Now an alternative method we could have used would have been to recall the general formula for the equation of a straight line. That’s 𝑦 equals 𝑚𝑥 plus 𝑐, where 𝑚 is the gradient and 𝑐 is the 𝑦-intercept. Comparing this to our equation, we can see that our graph has a 𝑦-intercept of three. The gradient is the coefficient of 𝑥; it’s the number of 𝑥s we have, which is just one.
A gradient of one means that, for every one square right we move on the graph, we move one square up. We could’ve started by plotting the 𝑦-intercept, which is three. The line crosses the 𝑦-axis at three. We said that a gradient of one means that, for every one square right we move, we move one square up. So we can draw these horizontal and vertical lines, one right, one up.
To get to the next coordinate, again, we move one right, one up. Notice how these points, as expected, lie on the line we already plotted. But if we hadn’t already plotted that line, we mustn’t forget to connect these with a nice straight line that passes from one side of our axes to the other. And there we have it. There’s the line 𝑦 equals 𝑥 plus three drawn on the grid.