Calculate the slope of any line by drawing a right triangle beneath the line. For example, consider a line that goes through the origin and makes a 45-degree angle with the X axis. This line goes through points where both coordinates are the same --- like (1,1) and (3,3) and (147,147). If we draw a right triangle below the line by dropping a line down from (3,3) and across from (1,1), the lines will meet at (1.3). To compute the slope, divide the height of the triangle by the base of the triangle. In this case, slope = 2/2 = 1.
Get the sign without calculating the slope or even drawing a triangle. The slope of a line is positive if Y values of points get larger as X values increase, and the slope of the line is negative if the Y values of points get smaller as the X values increase. The slope formula --- whether the coordinates are positive or negative --- is slope = (Y2 - Y1) / (X2 - X1), where (X1, Y1) and (X2, Y2) are the points on the line that you are using to construct the triangle.
Find the slope from negative coordinates the same way you find the slope of a line with the reference triangle entirely in the first quadrant. The only difference is you have to be careful with all the extra minus signs. For example, consider the line that runs through the points (X1, Y1) = (-3, -1) and (X2, Y2) = (-1, -4). If we use these coordinates to compute the slope, all the coordinates are negative. Just plug in the negative values and compute according to the formula: slope = (Y2 - Y1) / (X2 - X1) = ((-4) - (-1)) / ((-1) - (-3)) = (-4 + 1) / (1 - 3) = -3/2.