Quantitive Business Methods

Drawing Graphs

When you look at a straight-line graph, there are two obvious features: , the intercept, which shows where the line crosses the y-axis the gradient, which shows how steep the line is. When the line crosses the y-axis, x has the value 0. And if you substitute x = 0 into the equation y = ax + b, you see that ax = 0, so y = b. In other words, the constant b is the intercept of the line on the y-axis. The gradient of a line shows how quickly it is rising (or falling), and is defined as the change in y for a unit change in x. The gradient of a straight line is clearly the same at any point, so we can find the increase in y when x increases from, say, n to n + 1: when x = n, y = ax + b = an + b 1 when x = n + 1, y = ax + b = a(n + 1) + b = an + a + b. The difference between these two is a, and this shows that an increase of 1 in x gives an increase of a in y. So the constant a is the gradient, meaning that the general equation for a straight line is: y = gradient x x + intercept Draw a graph of y = 10x + 50. Solution This is a straight-line graph of the standard form y = ax + b, with a = 10 and b = 50. We need only two points to draw the line and can take any convenient ones. For instance, we can arbitrarily take the points where x = 0 and x = 20. When x = 0, y = 10x + 50 = 10 x 0 + 50 = 50, When x = 20, y = 10x + 50 = 10 x 20 + 50 = 250, which defines the point (0,50). which defines the point (20, 250). Worked example 3.2 continued y 300 250 (20, 250) 200 150 100 50 * (0, 50) 0 0 5 10 15 20 25 Figure 3.7 Straight-line graph of y = 10x + 50