How do you sketch the graph of y=(x-1)/(x+1)?

The first step in tackling this problem is to find the intersection points of the graph with the x and y axes. The y-intercept can be found by substituting x=0 into the function, so y=(0-1)/(0+1)=-1 and the y-intercept is (0, -1). The x-intercept (i.e. the roots of the equation) occur when y=0, which only happens when the numerator of the equation, x-1, is equal to zero. Hence we can see that the equation has only one root which is at (1, 0). The next step is to find the stationary points of the equation, which occur when the gradient is zero. The equation can be differentiated using the quotient rule (can be explained if needed) to find that dy/dx=2/(x+1)^2. We can see that the value of the gradient will never reach zero due to the constant value of 2 in the numerator, hence we can conclude that the equation has no stationary points.

The next step is to check if there are any asymptotes. Vertical asymptotes occur when y becomes undefined, which, in this case, refers to when the denominator of the function equals zero, so when x=-1. Horizontal asymptotes are found by letting x tend to infinity, so as x tends to positive infinity, y tends to 1; and when x tends to negative infinity, y also tends to 1. So finally with all of the information, the graph can be sketched to show a hyperbolic function consisting of two separate sections, with the lower-right section crossing both x and y axes, and upper-left section in the second quadrant which does not intersect the axes.