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 π β Β±β, π β?
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. π¦ = π₯3 β 4π₯
2. π¦ = β(π₯ β 2)2 (π₯ β 4)
3. π¦ = π₯3 β 2π₯2 β 4π₯ + 8
4. π¦ = (π₯ β 2)(π₯ + 4)3 (π₯ + 1)2
To which conics are the following
radical equations related to
π¦ = Β± ππ₯ β β π¦ = Β± β β π₯2 π¦ = Β± β β ππ₯2 π¦ = Β± β + ππ₯2
π¦ = Β± β β ππ₯
Example
1. π¦ = π₯
2. π¦ = β π₯ + 3 β 5
3. π¦ = π₯2 β 3π₯ β 4 β 5
4. π¦ = 4 β π₯ β 5
5. π¦ = π₯2 β 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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