8.1 Inverse and Joint Variations

8.1 Inverse and Joint Variations Direct Variation y = kx, k≠ 0 or y/x = k

Inverse Variation y = k, k ≠ 0 or xy = k x Joint Variation z = kxy quantity varies directly with the product of two or more quantities

8.2 Simple Rational Function Graphs Rational Function f(x) = p(x) , p(x) & q(x) are polynomials q(x) and q(x) ≠ 0

Parent Function: f(x) = a x hyperbola - 2 symmetrical branches Domain: all non zero real numbers Range: all non zero real numbers Vertical asymptote: x = 0 Horizontal asymptote: y = 0 g(x) = 1 , a≠ 0 same domain, range, & asymptotes x Translations: y = a + k asymptotes: (x-h) x = h & y = k

Other Rational Functions: y = ax + b cx + d

Vertical Asymptote: x = -d (solve the denominator) c

Horizontal Asymptote: y = a

C 8.3 Graphs of Rational Functions f(x) = p(x) q(x) p(x) and q(x) have no common factors other than ±1

Graph: 1. x-intercepts are real zeros of p(x) 2. Vertical asymptotes @ each real zero of q(x) 3. Horizontal asymptote @ most 1 determined by degree: m: numerator degree n: denominator degree m < n y = 0 m = n y = a lead coefficient - numerator c lead coefficient -denominator m > n no horizontal asymptote

4. Make a table on each side of each vertical asymptote

8.4 Simplify, Multiply & Divide Rational Expressions

Simplify Rational Expressions: 1. Factor numerator and denominator 2. Divide out like factors

Multiply or Divide: 1. For divide problems, write as multiply by reciprocal of the second fraction 2. Factor all numerators and denominators 3. Divide out all common factors 4. Multiply numerators and multiply denominators 8.5A Add & Subtract Rational Expressions Add/Subtract

1. Factor denominators 2. Find LCD - multiply the distinct factors to the highest power in any single denominator 3. Multiply each fraction to build each for a Common denominator 4. Add/Subtract numerators - keep LCD 5. Reduce

8.5B Simplify Complex Fractions: Method 1 1. Write numerator & denominator as single fraction - Horizontal division 2. Multiply by the reciprocal

Method 2 1. Multiply all terms by LCD ( minor denominators) 2. Simplify

8.6 Solve Rational Equations

Method 1 1. Cross Multiply if in proportion form. ( fraction = fraction) 2. Solve 3. Check

Method 2 1. Multiply every term by the LCD (eliminates denominators) 2. Solve - Use appropriate method 3. Check