Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Math 31 Pre-Calculus Name: __________________ Pre-Calculus Review Slope & Equations of Lines Recall: rise y y 2 y1 run x x2 x1 Slope formula: m Equation of a line: o Slope-intercept form: y mx b (m is the ________; b is the ______________) o Slope-point form: y y1 m( x x1 ) (m is the ________; x1 , y1 is a ________ ____________________) o General form: Ax By C 0 Class Example: Write the equation of the line through the points (5, -6) and (-3, 0) in general form. What is its slope? 1. State the slope of each linear relation: a) b) y 5 f ( x) 2 3x d) x 3 c) 4 x 6 y 5 0 2. Write the equation of each line in general form: 3 a) with a y-intercept of -6 and a slope of 5 1 3 ( x 2) 4 Math 31 Pre-Calculus b) through the point (-2, 4) and parallel to the line 2 x 4 y 8 0 c) through the points (-3, 5) and (4, -5) (If you need more practice on this topic, do p. 9 #1-3) Factoring Recall: Factoring Methods: - GCF - trinomial factoring (PSF, grid) - difference of squares The Factor Theorem: A polynomial P (x ) has x b as a factor if and only if P(b) 0 . Class Examples: Factor the following completely: a) 75 3x 2 b) 3n 2 17n 28 c) 2 x 3 13x 2 16 x 5 d) 3x 3 x 2 12 x 4 2 Math 31 Pre-Calculus NEW: Factoring a difference of cubes: a 3 b 3 Factoring a sum of cubes: a 3 b 3 Class Examples: Factor the following: a) x 3 27 b) 8 x 3 y y 4 You can factor expressions that involve fractional or negative exponents by factoring out a common factor with the lowest exponent. Class Examples: Factor the following completely: 3 1 a) x 2 10 x 2 25 x 12 b) 2 128 x 3 3. Factor the following completely: a) y 2 2 y 35 b) 49 x 2 9 y 2 c) n 4 16 d) 8 x 2 14 x 5 e) 49 x 1 x f) 216 x3 g) 27 x 3 8 h) n 3 n 3 6n 3 7 3 4 1 Math 31 Pre-Calculus i) 1 9 x 1 20 x 2 k) x 2 1 2 2 6 x2 2 j) x 4 5 x3 x 5 12 l) 3x 3 10 x 2 x 12 (If you need more practice on this topic, do p. 3 #1-4) Rational Expressions Recall: Always factor before simplifying. You can never cancel half a binomial. Adding and subtracting rational expressions requires a _____________ _____________________. Non-permissible values are any that make the _____________________ equal zero. Class Examples: Simplify the following: 3n 9 a) 3 n 27 b) 4 2 x 1 x x2 Math 31 Pre-Calculus Perform the indicated operation. State the non-permissible values. 12 x 3x 4 x2 1 a) 2 b) x 1 x 1 x4 x 4. Simplify the following: x 2 3x a) x4 b) 1 5 x 1 c) x3 d) 5 x 2 5x 6 x2 4 x 2 4x 3 1 1 x2 Math 31 Pre-Calculus 5. Perform the indicated operation. State the non-permissible values. 2 5y4 1 x2 a) 3 b) x 2 2x 3 8 y c) x 4 2x 1 x d) 2 3 x 1 2x 2 2 Composition of Functions Recall: In a composite function f g (x) or f g , the function g (x ) replaces the variable in f (x) . Class Example: For the following functions, find f g , g f , f f , and g g . 1 f ( x) , g ( x) x x4 6 Math 31 Pre-Calculus 6. For the following functions, find f g , g f , f f , and g g . a) f ( x) x 2 , g ( x) 6 x b) f ( x) x , g ( x) 9 x 2 c) f (x) 4 x 1 , g ( x ) (omit f f ) 2 x x 1 (If you need more practice on this topic, do p. 67 #1-2) 7 Math 31 Pre-Calculus Rationalizing Recall: Rationalizing a denominator means to write an equivalent expression without a radical in the denominator. (We can also rationalize numerators.) We do this by multiplying the numerator and denominator by the _________________ _________________ e.g. x a x a x 2 a Sometimes this allows us to simplify the rational expression. Class Examples: Rationalize the denominator and simplify Rationalize the numerator and simplify x 2 16 x 2 . 7 x2 2 . x3 7. Rationalize the denominator and simplify: x 5 a) x 5 b) 8 4x 2 x 2 Math 31 Pre-Calculus 8. Rationalize the numerator and simplify: 3 x 5 x 3 a) b) x 2 16 x 9 (If you need more practice on this topic, do p. 4 #1-2) Once you can do all the math in this booklet, you should be ready to start calculus! 9