Polynomial

P(x) = a n x n + a n–1 x n– a 1 x + a 0, a n  0 1. a n > 0 and n even Graph of P(x) increases without bound as x decreases to the left and as x increases to the right. P(x)  ∞ as x  – ∞ P(x)  ∞ as x  ∞ 2. a n > 0 and n odd Graph of P(x) decreases without bound as x decreases to the left and increases without bound as x increases to the right. P(x)  – ∞ as x  – ∞ P(x)  ∞ as x  ∞ Left and Right Behavior of a Polynomial Polynomial

Left and Right Behavior of a Polynomial P(x) = a n x n + a n–1 x n– a 1 x + a 0, a n  0 3. a n < 0 and n even Graph of P(x) decreases without bound as x decreases to the left and as x increases to the right. P(x)  – ∞ as x  – ∞ P(x)  – ∞ as x  ∞ 4. a n < 0 and n odd Graph of P(x) increases without bound as x decreases to the left and decreases without bound as x increases to the right. P(x)  ∞ as x  – ∞ P(x)  – ∞ as x  ∞

Any proper fraction P(x)/D(x) reduced to lowest terms can be decomposed in the sum of partial fractions as follows: 1. If D(x) has a nonrepeating linear factor of the form ax + b, then the partial fraction decomposition of P(x)/D(x) contains a term of the form A a constant 2. If D(x) has a k-repeating linear factor of the form (ax + b) k, then the partial fraction decomposition of P(x)/D(x) contains k terms of the form 3. If D(x) has a nonrepeating quadratic factor of the form ax 2 + bx + c, which is prime relative to the real numbers, then the partial fraction decomposition of P(x)/D(x) contains a term of the form 4. If D(x) has a k-repeating quadratic factor of the form (ax 2 + bx + c) k, where ax 2 + bx + c is prime relative to the real numbers, then the partial fraction decomposition of P(x)/D(x) contains k terms of the form A ax + b A 1 + b + A 2 ( + b ) 2 + … + A k ( ax + b ) k A 1, A 2, …, A k constants Ax + B ax 2 + bx + c A, B constants A 1 x + B 1 ax 2 + bx + c + A 2 x + B 2 ( ax 2 + bx + c ) 2 + … + A k x + B k ( ax 2 + bx + c ) k A 1, …, A k, B 1 B k constants Partial Fraction Decomposition

Circle Ellipse Parabola Hyperbola Conic Sections

1. y 2 = 4ax Vertex: (0, 0) Focus: (a, 0) Directrix: x = –a Symmetric with respect to the x axis. Axis the x axis 2. x 2 = 4ay Vertex: (0, 0) Focus: (0, a) Directrix: y = –a Symmetric with respect to the y axis. Axis the y axis a 0 (opens right) a 0 (opens up) Standard Equations of a Parabola with Vertex at (0, 0)

Standard Equations of an Ellipse with Center at (0, 0) [Note: Both graphs are symmetric with respect to the x axis, y axis, and origin. Also, the major axis is always longer than the minor axis.]

2. y 2 a 2 – x 2 b 2 = 1 x intercepts: none y intercepts: ± a (vertices) Foci: F' (0, – c ) F (0, c ) c 2 = a 2 + b 2 Transverse axis length = 2 a Conjugate axis length = 2 b 1. x 2 a 2 + y 2 b 2 = 1 x intercepts: ± a (vertices) y intercepts: none Foci: F' (– c, 0) F ( c c 2 = a 2 + b 2 Transverse axis length = 2 a Conjugate axis length = 2 b [Note: Both graphs are symmetric with respect to the x axis, y axis, and origin.] Standard Equations of a Hyperbola with Center at (0, 0)

Circles (x – h) 2 + (y – k) 2 = r 2 Center (h, k) Radius r (x – h) 2 = 4a(y – k) Vertex (h, k) Focus (h, k + a) a > 0 opens up a < 0 opens down (y – k) 2 = 4a(x – h) Vertex (h, k) Focus (h + a, k) a < 0 opens left a > 0 opens right Parabolas Standard Equations for Translated Conics—I

Ellipses Standard Equations for Translated Conics—II

Hyperbolas Standard Equations for Translated Conics—II

Contoh

Fungsi Implisit Bentuknya f(x,y ) = 0 dimana x dan y berada pada satu ruas kiri atau kanan Contoh yaitu ellips dengan setengah sumbu panjang dan pendek 2 dan 4

Koordinat Polar