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Curve Sketching of Polynomial in Factored Form
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 a
large numbers of points
required for a detailed plot.
Basic Techniques of Curve Sketching
Determine the x- and y- intercepts of the curve.
Determine the symmetry of the curve.
wrt the x-axis? y-axis? origin?
Determine the end behavior.
As ๐ โ ยฑโ, ๐ โ?
Determine the shape of the graph near a zero.
If the multiplicity of the zeros is odd, then the graph will cross the x-axis at the zeros. Otherwise, it will not cross the x-axis.
Examples
1. ๐ฆ = ๐ฅ3 โ 4๐ฅ
2. ๐ฆ = โ(๐ฅ โ 2)2 (๐ฅ โ 4)
3. ๐ฆ = ๐ฅ3 โ 2๐ฅ2 โ 4๐ฅ + 8
4. ๐ฆ = (๐ฅ โ 2)(๐ฅ + 4)3 (๐ฅ + 1)2
To which conics are the following
radical equations related to
๐ฆ = ยฑ ๐๐ฅ โ โ ๐ฆ = ยฑ โ โ ๐ฅ2 ๐ฆ = ยฑ โ โ ๐๐ฅ2 ๐ฆ = ยฑ โ + ๐๐ฅ2
๐ฆ = ยฑ โ โ ๐๐ฅ
Example
1. ๐ฆ = ๐ฅ
2. ๐ฆ = โ ๐ฅ + 3 โ 5
3. ๐ฆ = ๐ฅ2 โ 3๐ฅ โ 4 โ 5
4. ๐ฆ = 4 โ ๐ฅ โ 5
5. ๐ฆ = ๐ฅ2 โ 9
Sketch
1. y = (x-2)(x+4)2 (x+1)
2. y = (x-2)2(x+4)2 (x+1)
3. y = (x-2)(x+4) (x+1)2
4. y = (x-2)(x+4)3 (x+1)2
Write equation for
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