CURVE SKETCHING PRECALC1 (Analytical Geometry)

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 𝑦 = 𝑥3 − 4𝑥 𝑦 = −(𝑥−2)2 (𝑥−4) 𝑦 = 𝑥3 − 2𝑥2 − 4𝑥 + 8 𝑦 = (𝑥−2)(𝑥+4)3 (𝑥+1)2

𝑦 = 𝑥3 − 4𝑥

𝑦 = −(𝑥−2)2 (𝑥−4)

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− 𝑥 2 𝑦=± ℎ−𝑔 𝑥 2 𝑦=± ℎ+𝑔 𝑥 2 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 𝑦= 𝑥 𝑦=− 𝑥+3 −5 𝑦= 𝑥 2 −3𝑥−4 −5 𝑦= 4−𝑥 −5 𝑦= 𝑥 2 −9

Other Examples of Radical Function 𝑦= 3 𝑥 𝑦=− 3 𝑥+2 +5

HOMEWORK

Sketch Write equation for y = (x-2)(x+4)2 (x+1) y = (x-2)2(x+4)2 (x+1)