This page covers Curve Sketching within Coordinate Geometry.
Given a particular equation, you need be able to draw a quick sketch of its 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 knowledge 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 curve, there are a number of things that you should work out before drawing your sketch:
Asymptotes- these are lines for which the graph is undefined (this means that the curve does not cross asymptotes). 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. A curve often gets very close to an asymptote, without actually crossing it.
Work out 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 sketch.
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 small (large and negative)?
Is the graph symmetrical about the x or y-axes? Remember that the graph is symmetrical about the y-axis if replacing x by -x in the equation of the graph doesn't change the equation (for example y = x2 is symmetrical about the y-axis because if x is replaced by -x, the value of y is not changed since (-x)2 = x2). Functions which are symmetrical in the y-axis are known as even functions.
The graph is symmetrical about the x-axis if replacing y by -y does not change the equation of the graph. For example y2 = x.
The graph will have rotational symmetry if f(x) = -f(-x), in other words if replacing x by -x in the equation only results in the sign of the equation being changed. Such functions are known as odd functions.
You may also think about where the maxima and minima occur (by differentiating). These can then be marked onto your sketch.
Sketch the graph of y = 1 + x
1 - x
1) Asymptotes: 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 cannot happen, since -1 ¹ 1, so the graph cannot be defined for y = -1. This is therefore another asymptote.
2) Where the axes are crossed: 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 positive 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 axis.
The sketch of the graph would therefore look something like this:
Note that the curve does not cut the lines that we have found to be asymptotes, but it gets extremely close to them.