# 1.5.2 Mathematical graphs: How do you read them?

The coordinates of a point are always given in the form

(value along the *x*-axis, value along the *y*-axis).

Two values separated by a comma and enclosed in round brackets form a coordinate pair. Figure 22 shows how the values of the coordinates specify points in the different quadrants. (3,2) is a point in the first quadrant. Its position is specified by moving three units horizontally along the *x*-axis followed by two units vertically, parallel to the *y*-axis.

Similarly, in Figure 23, (−21) specifies a point in the second quadrant, (−3, −3) is a point in the third quadrant and (1, −2) is a point in the fourth. No two points on a graph share the same coordinates unless they are at exactly the same position.

Gradients for mathematical graphs, are calculated in the usual way by dividing vertical distance by horizontal distance.Figure 24 shows a straight-line graph with two points at the coordinates (1,1) and (3,4).

The vertical distance between the points is just the numerical difference between the two *y*-coordinates, which is 4−1=3. The horizontal distance is the difference between the two *x*-coordinates, which is 3−1=2. As the *x*-coordinate increases (from 1 to 3), the corresponding *y*-coordinate also increases (from 1 to 4):

Since an increase in the *x*-coordinate is matched by an increase in the *y*-coordinate, the graph has a positive gradient. A straight-line graph with a positive gradient always slopes upwards from left to right.

Now look at Figure 25. Here, the plotted coordinates are (1, 4) and (3,1). The vertical distance is still 3 and the horizontal distance is still 2, but the graph is now sloping down from left to right rather than up. This time, as the *x*-coordinate increases (from 1 to 3), the *y*-coordinate decreases (from 4 to 1).

Treating the change in the *y*-coordinate as a negative increase (as with bank accounts: a reduction in savings can be thought of as a negative increase in savings), write the change as a negative number:

The graph has a negative gradient. A straight-line graph with a negative gradient always slopes downwards from left to right.

## Negative gradients

In all the real-world examples so far, the gradient in the relationship formula has been positive. This is because in each case of conversions, increasing one quantity has resulted in an increase in the other (imagine getting *fewer* francs for *more pounds*!). There are situations, however, where*increasing*one quantity results in a decrease in the other, and conversely. In these cases, the constant in the formula is negative, as is the gradient of the corresponding straight line.

## Activity 10: Graph gradients

Points *A, B* and *C* are located at the coordinates *A* = (−6, 5), *B* = (2, −4) and *C* = (5,4).

Plot these points on a graph and calculate the gradients of the lines joining *A* to *B*, and *B* to *C*.

### Discussion

For the line joining *A* and *B*, the *x*-coordinate changes from −6 to +2, an increase of 8. The y-coordinate changes from 5 to −4, an ‘increase’ of −9. So the gradient is negative and equal to (−9)/8=−1.125, and the line slopes *down* from left to right.

For the line from *B* to *C*, the *x*-coordinate increases by 3, from 2 to 5. The y-coordinate increases by 8, from −4 to 4. So the gradient is positive and equal to 8/3(2.667), and the line slopes *up* from the left to right.