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Chapter 10

Chapter 10 . SWBAT solve problems using the Pythagorean Theorem. SWBAT perform operations with radical expressions. SWBAT graph square root functions. 10 – 1 The Pythagorean Theorem. Vocabulary: Hypotenuse Leg Pythagorean Theorem Conditional Hypotenuse Conclusion Converse.

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Chapter 10

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  1. Chapter 10 SWBAT solve problems using the Pythagorean Theorem.SWBAT perform operations with radical expressions. SWBAT graph square root functions.

  2. 10 – 1 The Pythagorean Theorem Vocabulary: Hypotenuse Leg Pythagorean Theorem Conditional Hypotenuse Conclusion Converse

  3. 10 – 1 The Pythagorean Theorem Vocabulary: Hypotenuse: The side opposite the right angle. Always the longest side Leg: Each side forming the right angle Pythagorean Theorem: relates the lengths of legs and length of hypotenuse Conditional: If-then statements Hypothesis: The part following the if Conclusion: The part following then Converse: Switches the hypothesis and conclusion

  4. 10-1 The Pythagorean Theorem

  5. Find the length of a Hypotenuse A triangle as leg lengths 6-inches. What is the length of the hypotenuse of the right triangle? a2 + b2 = c2 Pythagorean Theorem 62 + 62 = c2 Substitute 6 for a and b 72 = c2 Simplify √72 = c Find the PRINCIPAL square root 8.5 = c Use a Calculator The length of the hypotenuse is about 8.5 inches.

  6. You DO! What is the length of the hypotenuse of a right triangle with legs of lengths 9 cm and 12 cm? 15 cm

  7. Finding the Length of a Leg What is the side length b in the triangle below? a2 + b2 = c2 Pythagorean Theorem 52 + b2 = 132 Substitute 5 and 13 25 + b2 = 169 Simplify b2 = 144 Subtract 25 b = 12 Find the PRINCIPAL square root

  8. You Do! What is the side length a in the triangle below? 9

  9. The Converse of the Pythagorean Theorem

  10. Identifying a Right Triangle Which set of lengths could be the side lengths of a right triangle? • 6 in., 24 in., 25in. • 4m, 8m, 10m • 10in., 24in., 26in. • 8 ft, 15ft, 16ft

  11. Identifying a Right Triangle Which set of lengths could be the side lengths of a right triangle? Determine whether the lengths satisfy a2 + b2 = c2. The greatest length = c. • 6 in., 24 in., 25in. 62 + 242 =? 252 36 + 576 =? 625 612 ≠ 625

  12. Identifying a Right Triangle Which set of lengths could be the side lengths of a right triangle? Determine whether the lengths satisfy a2 + b2 = c2. The greatest length = c. b) 4m, 8m, 10m 42 + 82 =? 102 16 + 64 =? 100 80 ≠ 100

  13. Identifying a Right Triangle Which set of lengths could be the side lengths of a right triangle? Determine whether the lengths satisfy a2 + b2 = c2. The greatest length = c. c) 10in., 24in., 26in. 102 + 242 =? 262 100 + 576 =? 676 676 = 676

  14. Identifying a Right Triangle Which set of lengths could be the side lengths of a right triangle? Determine whether the lengths satisfy a2 + b2 = c2. The greatest length = c. By the Converse of the Pythagorean Theorem, the lengths 10 in., 24 in., and 26 in. could be the side lengths of a right triangle. The correct answer is C.

  15. You Do! Could the lengths 20 mm, 47 mm, and 52 mm be the side lengths of a right triangle? Explain. No; 202 + 472 ≠ 522

  16. Homework • Workbook Pages: pg. 288 1 – 31 odd

  17. 10-5 Graphing Square Root Functions Vocabulary: Square Root Function

  18. Square Root Functions A square root function is a function containing a square root with the independent variable in the radicand. The parent square root function is: y = √x . The table and graph below show the parent square root function.

  19. Essential Understanding • You can graph a square root function by plotting points or using a translation of the parent square root function. • For real numbers, the value of the radicand cannot be negative. So the domain of a square root function is limited to values of x for which the radicand is greater than or equal to 0.

  20. Finding the Domain of a Square Root Function • What is the domain of the function y = 2√(3x-9) ? 3x – 9 ≥ 0 The radicand cannot be negative 3x ≥ 9 Solve for x x ≥ 3 The domain of the function is the set of real numbers greater than or equal to 3.

  21. You Do! What is the domain of: y = x ≤ 2.5

  22. Graphing a Square Root Function Graph the function: I = ⅕√P, which gives the current I in amperes for a certain circuit with P watts of power. When will the current exceed 2 amperes? Step 1: Make a Table

  23. Graphing a Square Root Function Graph the function: I = ⅕√P, which gives the current I in amperes for a certain circuit with P watts of power. When will the current exceed 2 amperes? Step 1: Plot the points on a graph. The current will exceed 2 amperes when the power is more than 100 watts

  24. You Do! When will the current in the previous example exceed 1.5 amperes? 56.25 watts 1.5 = ⅕√P Substitute 1.5 for I 7.5 = √P Multiply by 5 (7.5)2 = (√P)2 Square both sides 56.25 = P Simplify

  25. Graphing a Vertical Translation For any number k, graphing y = (√x )+ k translates the graph of y = √x up k units. Graphing y = (√x)– k translates the graph of y = √x down k units.

  26. Graphing a Vertical Translation What is the graph of y = (√x) + 2 ?

  27. You Do! What is the graph of y = (√x) – 3 ?

  28. Graphing a Horizontal Translation For any positive number h, graphing y = translates the graph of y = √x to the left h units. Graphing y = translates the graph of y = √x to the right h units.

  29. Graphing a Horizontal Translation What is the graph of y = ?

  30. You Do! What is the graph of y =

  31. Homework Workbook pages 303-304 1 – 25 odd; 29*

  32. 10-2 Simplifying Radicals

  33. 1. Simplify Find a perfect square that goes into 147.

  34. 2. Simplify Find a perfect square that goes into 605.

  35. Simplify • . • . • . • .

  36. How do you simplify variables in the radical? What is the answer to ? Look at these examples and try to find the pattern… As a general rule, divide the exponent by two. The remainder stays in the radical.

  37. 4. Simplify Find a perfect square that goes into 49. 5. Simplify

  38. Simplify • 3x6 • 3x18 • 9x6 • 9x18

  39. 6. Simplify Multiply the radicals.

  40. 7. Simplify Multiply the coefficients and radicals.

  41. Simplify • . • . • . • .

  42. How do you know when a radical problem is done? • No radicals can be simplified.Example: • There are no fractions in the radical.Example: • There are no radicals in the denominator.Example:

  43. 8. Simplify. Divide the radicals. Uh oh… There is a radical in the denominator! Whew! It simplified!

  44. Uh oh… Another radical in the denominator! 9. Simplify Whew! It simplified again! I hope they all are like this!

  45. Uh oh… There is a fraction in the radical! 10. Simplify Since the fraction doesn’t reduce, split the radical up. How do I get rid of the radical in the denominator? Multiply by the denominator to make the denominator a perfect square!

  46. Homework Workbook Pages pg. 291 – 292 1 – 35 odd

  47. Chapter 11 OBJECTIVES: SWBAT to solve rational equations and proportions. SWBAT write and graph equations for inverse variations. SWBAT compare direct and inverse variations. SWBAT graph rational functions.

  48. 11-5 Solving Rational Equations Vocabulary: Rational Equation

  49. 11-5 Solving Rational Equations Vocabulary: A rational equation is an equation that contains one or more rational expression.

  50. Solving Equations With Rational Expressions What is the solution of (5/12) – (1/2x) = (1/3x)? (5/12) – (1/2x) = (1/3x) The denominators are 12, 2x, and 3x, so the LCD is 12x 12x[(5/12)-(1/2x)] = 12x(1/3x) Multiply by LCD 12x(5/12) – 12x(1/2x) = 12x( 1/3x) Dist. Prop. 5x – 6 = 4 Simplify 5x = 10 Solve for x x = 2 6 4

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