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Curve Sketching
Curve sketching is used to produce a rough idea of overall shape of a curve given its equation without computing a detailed plot.
Learning Objective

Use "curve sketching" to estimate a function's shape
Key Points
 Determine the
$x$  and$y$ intercepts of the curve.  Determine the symmetry of the curve.
 Determine any bounds on the values of
$x$ and$y$ .  Determine the asymptotes of the curve.
Terms

asymptote
a straight line which a curve approaches arbitrarily closely, as they go to infinity

symmetry
Exact correspondence on either side of a dividing line, plane, center or axis.
Full Text
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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Key Term Reference
 algebraic
 Appears in these related concepts: The Dot Product, Four Ways to Represent a Function, and Finding Limits Algebraically
 axis
 Appears in these related concepts: Area Between Curves, TwoComponent Forces, and Components of a Vector
 curve
 Appears in these related concepts: Integration By Parts, Arc Length and Surface Area, and Arc Length and Speed
 exponent
 Appears in these related concepts: Logarithmic Functions, Solving General Problems with Logarithms and Exponents, and Scientific Notation
 infinity
 Appears in these related concepts: Indeterminate Forms and L'Hôpital's Rule, Precise Definition of a Limit, and Horizontal Asymptotes and Limits at Infinity
 origin
 Appears in these related concepts: Adding and Subtracting Vectors Graphically, Types of Muscle Tissue, and ThreeDimensional Coordinate Systems
 tangent
 Appears in these related concepts: Derivatives of Exponential Functions, Direction Fields and Euler's Method, and Tangent Vectors and Normal Vectors
Sources
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Cite This Source
Source: Boundless. “Curve Sketching.” Boundless Calculus. Boundless, 20 May. 2016. Retrieved 25 May. 2016 from https://www.boundless.com/calculus/textbooks/boundlesscalculustextbook/derivativesandintegrals2/applicationsofdifferentiation10/curvesketching572788/