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Preliminaries

Preliminaries. 1. Precalculus Review I Precalculus Review II The Cartesian Coordinate System Straight Lines. 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.

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Preliminaries

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

  2. 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

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

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

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

  6. Absolute Value Notice the opposite sign To evaluate:

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

  8. Exponents n,m positive integers Definition Example n factors

  9. Laws of Exponents Law Example

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

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

  12. Polynomials • Multiplication Distribute Distribute Combine like terms

  13. Factoring Polynomials • Greatest Common Factor The terms have 6t2in common • Grouping Factor mx Factor –2

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

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

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

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

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

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

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

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

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

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

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

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

  26. The Distance Formula

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

  28. 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

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

  30. 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)

  31. 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. ).

  32. 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.

  33. 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 .

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

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

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

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

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