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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

symmetry
Exact correspondence on either side of a dividing line, plane, center or axis.

asymptote
a straight line which a curve approaches arbitrarily closely, as they go to infinity
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, Finding Limits Algebraically, and Four Ways to Represent a Function
 axis
 Appears in these related concepts: Area Between Curves, Regional Terms and Axes, 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: Logarithms of Products, Logarithms of Powers, and Logarithms of Quotients
 infinity
 Appears in these related concepts: Precise Definition of a Limit, Horizontal Asymptotes and Limits at Infinity, and Indeterminate Forms and L'Hôpital's Rule
 origin
 Appears in these related concepts: Types of Muscle Tissue, Lever Systems, and ThreeDimensional Coordinate Systems
 tangent
 Appears in these related concepts: Special Angles, Graphs of Exponential Functions, Base e, and Circular Motion
Sources
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Cite This Source
Source: Boundless. “Curve Sketching.” Boundless Calculus. Boundless, 26 May. 2016. Retrieved 26 Aug. 2016 from https://www.boundless.com/calculus/textbooks/boundlesscalculustextbook/derivativesandintegrals2/applicationsofdifferentiation10/curvesketching572788/