Preliminaries 1 Precalculus Review II The Cartesian Coordinate

Preliminaries 1 • Precalculus Review II • The Cartesian Coordinate System • Straight Lines 1

The Real Numbers The real numbers can be ordered and represented in order on a number line 0 -1. 87 4. 55 x -3 -2 -1 0 1 2 3 4 2

Inequalities, graphs, and notations Inequality Graph Interval ( ] 3 7 ( 5 ] ) or ( means not included in the solution ] or [ means included in the solution 3

Intervals Interval Graph Example (a, b) a b [a, b] a [ b ] (a, b] a ( b ] [a, b) a b (a, ) a ( [ ( (- , b) [a, ) (- , b] a ) ) b ) [ (3, 5) [4, 7] (-1, 3] [-2, 0) (1, ] ) (- , -3] 5 ( ) 4 7 [ ] -1 3 ( ] -2 0 [ ) 1 ( (- , 2) [0, b ) 3 2 ) 0 [ -3 ] 4

Properties of Inequalities If a, b, and c are any real numbers, then Property 1 Example 2 < 3 and 3 < 8, so 2 < 8. Property 2 Property 3 Property 4 5

Absolute Value To evaluate: Notice the opposite sign 6

Absolute Value Properties If a and b are any real numbers, then Example Property 5 Property 6 Property 7 Property 8 7

Exponents n, m positive integers Definition Example n factors 8

Laws of Exponents Law Example 9

Algebraic Expressions • Polynomials • Rational Expressions • Other Algebraic Fractions 10

Polynomials • Addition Combine like terms • Subtraction Distribute Combine like terms 11

Polynomials • Multiplication Distribute Combine like terms 12

Factoring Polynomials • Greatest Common Factor The terms have 6 t 2 in common • Grouping Factor mx Factor – 2 13

Factoring Polynomials • Difference of Two Squares: Ex. • Sum/Difference of Two Cubes: Ex. 14

Factoring Polynomials • Trinomials Ex. Trial and Error Ex. Greatest Common Factor Trial and Error 15

Roots of Polynomials • Finding roots by factoring (find where the polynomial = 0) Ex. 16

Roots of Polynomials • Finding roots by the Quadratic Formula • The Quadratic Formula: If with a, b, and c real numbers, then 17

Example Using the Quadratic Formula: Ex. Find the roots of Note values Here a = 3, b = 7, and c = 1 Plug in Simplify 18

Rational Expressions Operation Addition Subtraction Multiplication Division P, Q, R, and S are polynomials Notice the common denominator Find the reciprocal and multiply 19

Rational Expressions • Simplifying • Multiplying Factor Cancel common factors 2 Multiply Across 20

Rational Expressions • Adding/Subtracting Must have LCD: x(x + 4) Distribute and combine fractions Combine like terms 21

Other Algebraic Fractions • Complex Fractions Multiply by the LCD: x Distribute and reduce to get here Factor to get here 22

Other Algebraic Fractions Notice: • Rationalizing a Denominator Multiply by the conjugate Simplify 23

Cartesian Coordinate System y-axis (x, y) x-axis 24

Cartesian Coordinate System Ex. Plot (4, 2) Ex. Plot (-2, -1) Ex. Plot (2, -3) (4, 2) (-2, -1) (2, -3) 25

The Distance Formula 26

The Distance Formula Ex. Find the distance between (7, 5) and (-3, -2) 7 10 27

The Equation of a Circle A circle with center (h, k) and radius of length r can be expressed in the form: Ex. Find an equation of the circle with center at (4, 0) and radius of length 3 28

Straight Lines • Slope • Point-Slope Form • Slope-Intercept Form 29

Slope – the slope of a non-vertical line that passes through the points is given by: and Ex. Find the slope of the line that passes through the points (4, 0) and (6, -3) 30

Slope Two lines are parallel if and only if their slopes are equal or both undefined Two lines are perpendicular if and only if the product of their slopes is – 1. That is, one slope is the negative reciprocal of the other slope (ex. ). 31

Point-Slope Form An equation of a line that passes through the point with slope m is given by: Ex. Find an equation of the line that passes through (3, 1) and has slope m = 4. 32

Slope-Intercept Form An equation of a line with slope m and y-intercept is given by: Ex. Find an equation of the line that passes through (0, -4) and has slope . 33

Vertical Lines Can be expressed in the form x = a y x=3 x 34

Horizontal Lines Can be expressed in the form y = b y y=2 x 35

Example Find an equation of the line that passes through (-2, 1) and is perpendicular to the line Solution: Step 1. Step 2. 36

Example Find an equation of the line that passes through (0, 1) and is parallel to the line Solution: Step 1. Step 2. 37