What are the signs of the 𝑥- and 𝑦-coordinates of any point belonging to the quadrant shaded in the figure below?
If we look carefully at the figure, we can see that it shows the coordinate plane. The coordinate plane we could call a surface or a grid. And it allows us to plot different points. And these points can be in one of four quadrants. One, two — will come back to this one because this is the shaded one — three, four. And this problem is encouraging us to think about the quadrant that’s been shaded. This is the lower right quadrant.
We know that any point on the coordinate plane can be shown by writing a pair of coordinates. The first number is the 𝑥-coordinate. This shows us where we move along on the 𝑥-axis. And the second number is the 𝑦-coordinate. And this shows us where to move up and down on the 𝑦-axis. And the question asks us, what are the signs of the 𝑥- and 𝑦-coordinates of any point belonging to the shaded quadrant?
The phrase “what are their signs?” is really encouraging us to think about whether or not our number is going to be a positive number or a negative number. Now the 𝑥- and 𝑦-axes are simply number lines, just going in different directions. So to help us answer the question, let’s put some numbers on our number lines.
The point where the 𝑥- and the 𝑦-axes cross is the origin. This has a value of zero. And we start counting from this point. Now if we think about the 𝑥-axis first, this is a horizontal axis. And just like any horizontal number line, as we count to the right, we’re counting upwards one, two, three, four, and so on. These are positive numbers. And if we count to the left from zero, we’re going into the negative numbers.
And so what can we say about the sign of the 𝑥-coordinate of any point in this shaded quadrant? Is it going to be positive or negative? In this quadrant, the 𝑥-coordinate will always be positive. To help us think about the 𝑦-coordinate, let’s think about the 𝑦-axis as being a number line that goes up and down. It’s a vertical number line. As we move up the number line, we count forwards. These are the positive numbers. And if we start from zero and count down the number line, we count backwards. These are our negative numbers.
If we were to pick any point in this shaded quadrant, what sign would we expect the 𝑦-coordinate to be, positive or negative? If we look at the lower part of the 𝑦-axis, we can see that these are negative numbers. And so what are the signs of the 𝑥- and 𝑦-coordinates of any point belonging to the shaded quadrant? The 𝑥-coordinate will be positive, and the 𝑦-coordinate will be negative.