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

  • Sketching of Radical Equations

  • 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

  • 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