When you are asked to sketch a curve, you need be able to draw a quick sketch of the curve, showing the main details (such as where the curve crosses the axes). You should be able to quickly sketch straight line graphs, from your knowlege that in the equation y = mx + c, m is the gradient and c where the graph crosses the y-axis.
When asked to sketch a more complicated graph, there are a number of things that you should work out before drawing your sketch.
1) Asymptotes- these are lines for which the graph is undefined. Remember that you cannot divide by zero. Therefore, in the graph of 1/(1 + x), x = -1 is an asymptote because when x is -1, you end up dividing by zero.
2) Where the graph crosses the axes. The graph will cross the x-axis when y = 0 and the y-axis when x = 0. Substitute in x = 0 and then y = 0 to determine the crossing points, and mark these on your graph.
3) What happens as x becomes very large? Think about whether y will become very large, very small, positive or negative. What happens as x becomes very large and negative?
4) Is the graph symmetrical about the x or y axes? Remember, the graph is symmetrical about the y-axis if replacing x by -x in the equation of the graph doesn't change the equation. The graph is symmetrical about the x-axis if replacing x by -x does not change the equation of the graph, apart from making the equation the negative of the original equation.
5) You may also think about where the maxima and minima occur (by differentiating).
Example:
Sketch the graph of y = 1 + x
1 - x
1) When x = 1, we end up dividing by zero so there will be an asymptote at x = 1.
Also think about what happens when y = -1.
-1 = 1 + x
1 - x
-1(1 - x) = 1 + x
-1 + x = 1 + x
-1 = 1.
This is clearly wrong, so the graph cannot be defined for y = -1. This is therefore another asymptote.
2) When x = 0, y = 1. Therefore the curve crosses the y-axis at (0,1).
When y = 0, 1 + x = 0 so x = -1. Therefore the curve crosses the x-axis at (-1, 0).
3) As x becomes large, 1 + x will become large and 1 - x will become large and negative. Therefore as x becomes large, y = large/-large = -1. As x becomes very large and negative, 1 + x will become very large and negative and 1 - x will become very large and positive. Therefore y = -large/large = -1.
4) By substituting in -x for x it can be seen that the graph is not symmetrical in the x or y axes.
