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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 𝒙 → ±∞, 𝒚 →?
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. 𝑦 = 𝑥3 − 4𝑥
2. 𝑦 = −(𝑥 − 2)2 (𝑥 − 4)
3. 𝑦 = 𝑥3 − 2𝑥2 − 4𝑥 + 8
4. 𝑦 = (𝑥 − 2)(𝑥 + 4)3 (𝑥 + 1)2
To which conics are the following
radical equations related to
𝑦 = ± 𝑔𝑥 − ℎ 𝑦 = ± ℎ − 𝑥2 𝑦 = ± ℎ − 𝑔𝑥2 𝑦 = ± ℎ + 𝑔𝑥2
𝑦 = ± ℎ − 𝑔𝑥
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