TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Curve Sketching
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 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 graph, 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. Remember that you cannot divide by zero. Therefore, in the graph of y = 1/(1 + x), x = -1 is an asymptote because when x is -1, you end up dividing by zero. What happens as the curve approaches an asymptote? Which way does it approach from?
- 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.
- 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?
- 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 thegraph 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.
- You may also think about where the maxima and minima occur (by differentiating).
Example
Sketch 
- When x = 1, we end up dividing by zero so there will be an asymptote at x = 1.
We can also rearrange this equation to get:
This shows that there is also an asymptote at y = -1.
- 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).
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(which can be easily verified using polynomial division). As x gets large and positive, the fraction will become small and negative (2 divided by a large, negative number), so y tends towards -1 (from the 'bottom'). As x gets large and negative, the fraction will become small and positive (2 divided by a large, positive number), so y tends towards -1 (from the 'top').
(which can again be easily verified). As y gets large and positive, x tends to 1 from the left; as y gets large and negative, x tends to 1 from the right.
- By substituting in -x for x it can be seen that the graph is not symmetrical in the x or y axes.
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