CURVE SKETCHING - FCAMPENA...Curve Sketching of Polynomial in Factored Form In geometry, curve sketching (or curve tracing) includes techniques that can be used to produce a rough

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 = −3𝑥 + 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