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In geometry, curve sketching (or curve tracing) includes techniques that can be used to produce a rough idea of overall shape of a plane curve given its equation without computing the large numbers of points required for a detailed plot. It is an application of the theory of curves to find their main features.
The following steps are usually easy to carry out and give important clues as to the shape of a curve:
Determine the $x$- and $y$-intercepts of the curve. The $x$-intercepts are found by setting $y$ equal to $0$ in the equation of the curve and solving for $x$. Similarly, the y intercepts are found by setting $x$ equal to $0$ in the equation of the curve and solving for $y$.
Determine the symmetry of the curve. If the exponent of $x$ is always even in the equation of the curve, then the $y$-axis is an axis of symmetry for the curve. Similarly, if the exponent of $y$ is always even in the equation of the curve, then the $x$-axis is an axis of symmetry for the curve. If the sum of the degrees of $x$ and $y$ in each term is always even or always odd, then the curve is symmetric about the origin and the origin is called a center of the curve.
Determine any bounds on the values of $x$ and $y$. If the curve passes through the origin then determine the tangent lines there. For algebraic curves, this can be done by removing all but the terms of lowest order from the equation and solving. Similarly, removing all but the terms of highest order from the equation and solving gives the points where the curve meets the line at infinity.
Determine the asymptotes of the curve. Also determine from which side the curve approaches the asymptotes and where the asymptotes intersect the curve.
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