     # CURVE SKETCHING - FCAMPENA ... Curve Sketching of Polynomial in Factored Form In geometry, curve sketching

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• PRECALC1 (Analytical Geometry)

CURVE SKETCHING

• Curve Sketching of Polynomial Functions in Factored Form

• 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

• 𝑦 = 𝑥3 − 4𝑥

• 𝑦 = −(𝑥 − 2)2 (𝑥 − 4)

• 1. 𝑦 = 𝑥3 − 2𝑥2 − 4𝑥 + 8

• 𝑦 = 𝑥 − 2 𝑥 + 4 3 𝑥 + 1 2

• To which conics are the following

𝑦 = ± 𝑔𝑥 − ℎ 𝑦 = ± ℎ − 𝑥2 𝑦 = ± ℎ − 𝑔𝑥2 𝑦 = ± ℎ + 𝑔𝑥2

𝑦 = ± ℎ − 𝑔𝑥

• Example

𝑦 = 𝑥

• Example

𝑦 = 𝑥

• Example2:

𝑦 = − 𝑥 + 3 − 5

• Example2:

𝑦 = − 𝑥 + 3 − 5

• Example2:

𝑦 = − 𝑥 + 3 − 5

• Example2:

𝑦 = − 𝑥 + 3 − 5

• Example2:

𝑦 = − 𝑥 + 3 − 5

• Example

1. 𝑦 = 𝑥

2. 𝑦 = − 𝑥 + 3 − 5

3. 𝑦 = 𝑥2 − 3𝑥 − 4 − 5

4. 𝑦 = 4 − 𝑥 − 5

5. 𝑦 = 𝑥2 − 9

• Other Examples of Radical Function

1. 𝑦 = 3 𝑥

2. 𝑦 = − 3 𝑥 + 2 + 5

• HOMEWORK

• 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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