Chapter 1: Precalculus Review

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1 : Precalculus Review Math January 2018

2 Overview 1 Important Notation 2 Exponents 3 Polynomials 4 Rational Functions 5 Cartesian Coordinates 6 Lines

3 Notation Intervals: Interval Notation (a, b) (a, b] [a, b) [a, b] (, a] (, a) [a, ) (a, ) Corresponding Inequality All x such that a < x < b All x such that a < x b All x such that a x < b All x such that a x b All x such that x a All x such that x < a All x such that a x All x such that a < x

4 Notation A set is a collection of things. Intervals are an example of sets. Another example is to say that the United States is a set of states. We denote sets with {} e.g. {a, b, c} or {x R : a < x < b} Set Notation Read as... In English A B A intersect B What s in both A and B? A B A union B What s in A,B, or both? x A x is an element of A x is an object in A.

5 Examples If A = (0, 1) = {x R : 0 < x < 1}, B = [ 1, 1] = {x R : 1 x 1}, and C = ( 1 2, 3 2 ) = {x R : 1 2 < x < 3 2 }, then we have: A C = ( 1 2, 1) A B = (0, 1) B C = [ 1, 3 2 ) 1 2 A

6 Exponents Laws of Exponents a m a n = a m+n a m a n = a m n for a 0 (a m ) n = a m n (ab) n = a n b n CAUTION:In general (a + b) n a n + b n. Try a = b = 1 to see an example.

7 Examples Example 1: Simplify 4 16x 4 y 8.

8 Examples Example 1: Simplify 4 16x 4 y x 4 y 8 = (16x 4 y 8 ) 1 4

9 Examples Example 1: Simplify 4 16x 4 y x 4 y 8 = (16x 4 y 8 ) 1 4 = x 4 4 y 8 4

10 Examples Example 1: Simplify 4 16x 4 y x 4 y 8 = (16x 4 y 8 ) 1 4 = x 4 4 y 8 4 = 2xy 2 Example 2: Rationalize 3 2 x.

11 Examples Example 1: Simplify 4 16x 4 y x 4 y 8 = (16x 4 y 8 ) 1 4 = x 4 4 y 8 4 = 2xy 2 Example 2: Rationalize x = 3 2 x x x 2 x.

12 Examples Example 1: Simplify 4 16x 4 y x 4 y 8 = (16x 4 y 8 ) 1 4 = x 4 4 y 8 4 = 2xy 2 Example 2: Rationalize x = 3 2 x x x = 3 x 2x 2 x. Example 3: Expand (a + b) 2.

13 Examples Example 1: Simplify 4 16x 4 y x 4 y 8 = (16x 4 y 8 ) 1 4 = x 4 4 y 8 4 = 2xy 2 Example 2: Rationalize x = 3 2 x x x = 3 x 2x 2 x. Example 3: Expand (a + b) 2. (a + b) 2 = (a + b)(a + b)

14 Examples Example 1: Simplify 4 16x 4 y x 4 y 8 = (16x 4 y 8 ) 1 4 = x 4 4 y 8 4 = 2xy 2 Example 2: Rationalize x = 3 2 x x x = 3 x 2x 2 x. Example 3: Expand (a + b) 2. (a + b) 2 = (a + b)(a + b) = a 2 + ab + ab + b 2

15 Examples Example 1: Simplify 4 16x 4 y x 4 y 8 = (16x 4 y 8 ) 1 4 = x 4 4 y 8 4 = 2xy 2 Example 2: Rationalize x = 3 2 x x x = 3 x 2x 2 x. Example 3: Expand (a + b) 2. (a + b) 2 = (a + b)(a + b) = a 2 + ab + ab + b 2 = a 2 + 2ab + b 2

16 Polynomials Some Useful Factoring Formulas x 2 y 2 = (x + y)(x y) x 2 ± 2xy + y 2 = (x ± y) 2 x 3 ± y 3 = (x ± y)(x 2 xy + y 2 ) Examples: x = (x + 3)(x 2 3x + 9) x 2 1 = (x 1)(x + 1) x 2 2x + 1 = (x 1) 2

17 Quadratic Formula Theorem (The Quadratic Formula) The solutions of the equation ax 2 + bx + c = 0 with (a 0) are given by x = b ± b 2 4ac. 2a

18 Simplifying Rational Functions To simplify rational functions you need to factor the numerator and denominator then cancel common factors. Example: Simplify x2 x 6 x 2 4

19 Simplifying Rational Functions To simplify rational functions you need to factor the numerator and denominator then cancel common factors. Example: Simplify x2 x 6 x 2 4 x 2 x 6 x 2 4

20 Simplifying Rational Functions To simplify rational functions you need to factor the numerator and denominator then cancel common factors. Example: Simplify x2 x 6 x 2 4 x 2 x 6 x 2 4 = (x 3)(x+2) x 2 4

21 Simplifying Rational Functions To simplify rational functions you need to factor the numerator and denominator then cancel common factors. Example: Simplify x2 x 6 x 2 4 x 2 x 6 x 2 4 = (x 3)(x+2) x 2 4 = (x 3)(x+2) (x 2)(x+2)

22 Simplifying Rational Functions To simplify rational functions you need to factor the numerator and denominator then cancel common factors. Example: Simplify x2 x 6 x 2 4 = (x 3)(x+2) (x 2)(x+2) = x 3 x+2 x 2 x 6 x 2 4 = (x 3)(x+2) x 2 4

23 Rationalizing Algebraic Expressions Many times we want to remove any square roots from our denominator (numerator). To do so we will rationalize our denominator (numerator). Example: Rationalize the denominator of 1 1+ x

24 Rationalizing Algebraic Expressions Many times we want to remove any square roots from our denominator (numerator). To do so we will rationalize our denominator (numerator). Example: Rationalize the denominator of 1 1+ x 1 1+ x

25 Rationalizing Algebraic Expressions Many times we want to remove any square roots from our denominator (numerator). To do so we will rationalize our denominator (numerator). 1 Example: Rationalize the denominator of 1+ x 1 1+ x = 1 1+ x 1 x 1 x

26 Rationalizing Algebraic Expressions Many times we want to remove any square roots from our denominator (numerator). To do so we will rationalize our denominator (numerator). Example: Rationalize the denominator of x = 1 1+ x 1 x 1 x = 1 x 1 ( x) 2 1+ x

27 Rationalizing Algebraic Expressions Many times we want to remove any square roots from our denominator (numerator). To do so we will rationalize our denominator (numerator). Example: Rationalize the denominator of x = 1 1+ x 1 x 1 x = 1 x 1 ( = 1 x x) 2 1 x 1+ x

28 Basics We will denote a point on the Cartesian plane by (x, y). The plane is divided into four quadrants separated by the x-axis and the y-axis. Theorem (The Distance Formula) The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2.

29 Circles Using the distance formula we can derive the equation for a circle on the plane of radius r centered at (h, k) by noting that the circle will contain all points distance r from (h, k). Theorem (The Equation of a Circle) An equation for the circle centered at (h, k) with radius r is given by (x h) 2 + (y k) 2 = r 2.

30 Basics Theorem (Slope of a Line) If (x 1, y 1 ) and (x 2, y 2 ) lie on a line L, then the slope m of L is given by m = y x = y 2 y 1. x 2 x 1 If x 1 = x 2, then the slope is undefined and we have a vertical line. Make note that the slope can be thought of as the rate of change of y with respect to x, i.e, how fast y will change when x changes.

31 Basics The point where a line L intercepts the y-axis is called the y-intercept and looks like (0, b). The point where a line L intercepts the x-axis is called the x-intercept and looks like (a, 0). We call two line parallel if their slopes are equal or their slopes are both undefined. Two lines with slope m 1 and m 2 are perpendicular if and only if m 1 = 1 m 2.

32 Line Equations Theorem (Point-Slope Form) An equation of the line that has slope m and passes through the point (x 1, y 1 ) is given by y y 1 = m(x x 1 ). Theorem (Slope-Intercept Form) An equation of the line that has slope m and has a y-intercept at (0, b) is given by y = mx + b.

33 Line Equations Theorem (General Form) The equation Ax + By + C = 0 where A and B are not both zero is called the general form of a line. I will accept either of these three forms on the written homework. Keep in mind that depending on the information given there is probably one form or the other that is best to solve a given problem.

2-7 Solving Quadratic Inequalities. ax 2 + bx + c > 0 (a 0)

2-7 Solving Quadratic Inequalities. ax 2 + bx + c > 0 (a 0) Quadratic Inequalities In One Variable LOOKS LIKE a quadratic equation but Doesn t have an equal sign (=) Has an inequality sign (>,