Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Preliminaries 1 Precalculus Review I Precalculus Review II The Cartesian Coordinate System Straight Lines

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. The Real Numbers The real numbers can be ordered and represented in order on a number line x

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. InequalityGraphInterval ] ( ] ( ) or ( means not included in the solution ] or [ means included in the solution Inequalities, graphs, and notations

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Intervals IntervalGraph ( ) [ ] ( ] [ ) ( ) [ ] a b Example (a, b) [a, b] (a, b] [a, b) (a, ) (-, b) [a, ) (-, b] (3, 5) [4, 7] (-1, 3] [-2, 0) (1, ) (-, 2) [0, ) (-, -3] ( ) [ ] ( ] [ ) ( ) [ ] a b a a b b

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Properties of Inequalities If a, b, and c are any real numbers, then Property 1 Property 2 Property 3 Property 4 Example 2 < 3 and 3 < 8, so 2 < 8.

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Absolute Value To evaluate: Notice the opposite sign

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Absolute Value Properties If a and b are any real numbers, then Property 5 Property 6 Property 7 Property 8 Example

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Exponents Definition n factors Example n,m positive integers

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Laws of Exponents LawExample

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Algebraic Expressions Polynomials Rational Expressions Other Algebraic Fractions

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Polynomials Addition Combine like terms Subtraction Combine like terms Distribute

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Polynomials Multiplication Combine like terms Distribute

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Factoring Polynomials Greatest Common Factor Grouping The terms have 6t 2 in common Factor mxFactor –2

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Factoring Polynomials Sum/Difference of Two Cubes: Difference of Two Squares: Ex.

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Factoring Polynomials Trinomials Ex. Trial and Error Greatest Common Factor

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Roots of Polynomials Finding roots by factoring (find where the polynomial = 0) Ex.

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Roots of Polynomials The Quadratic Formula: If Finding roots by the Quadratic Formula with a, b, and c real numbers, then

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Example Using the Quadratic Formula : Ex. Find the roots of Here a = 3, b = 7, and c = 1 Plug in Note values Simplify

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Rational Expressions P, Q, R, and S are polynomials Addition Operation Multiplication Subtraction Division Notice the common denominator Find the reciprocal and multiply

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Rational Expressions Simplifying Cancel common factors Factor Multiplying Factor Cancel common factors 2 Multiply Across

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Rational Expressions Adding/Subtracting Combine like terms Must have LCD: x(x + 4) Distribute and combine fractions

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Other Algebraic Fractions Complex Fractions Factor to get here Distribute and reduce to get here Multiply by the LCD: x

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Other Algebraic Fractions Rationalizing a Denominator Simplify Multiply by the conjugate Notice:

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Cartesian Coordinate System y-axis x-axis (x, y)

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Cartesian Coordinate System Ex. Plot (4, 2) (4, 2) Ex. Plot (-2, -1) Ex. Plot (2, -3) (2, -3) (-2, -1)

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. The Distance Formula

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. The Distance Formula Ex. Find the distance between (7, 5) and (-3, -2) 7 10

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. 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

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Straight Lines Slope Point-Slope Form Slope-Intercept Form

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. 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)

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. 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. ).

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. 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.

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. 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.

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Vertical Lines x = 3 Can be expressed in the form x = a x y

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Horizontal Lines y = 2 Can be expressed in the form y = b x y

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Example Find an equation of the line that passes through (-2, 1) and is perpendicular to the line Solution: Step 1. Step 2.

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Example Find an equation of the line that passes through (0, 1) and is parallel to the line Solution: Step 1. Step 2.