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LESSON 2–1

LESSON 2–1. Power and Radical Functions. A. B. C. D. Describe the end behavior of f ( x ) = 4 x 4 + 2 x – 8. 5–Minute Check 2. C > 1 expands it 0 < C < 1 compresses it.

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LESSON 2–1

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  1. LESSON 2–1 Power and Radical Functions

  2. A. B. C. D. Describe the end behavior of f(x) = 4x4 + 2x – 8. 5–Minute Check 2

  3. C > 1 expands it 0 < C < 1 compresses it Identify the parent function f(x) of g(x) = 2|x – 3| + 1. Describe how the graphs of g(x) and f(x) are related. A.f(x) = | x |; f(x) is translated 3 units right, 1 unit up and expanded vertically to graph g(x). B.f(x) = | x |; f(x) is translated 3 units right, 1 unit up and expanded horizontally to graph g(x). C.f(x) = | x |; f(x) is translated 3 units left, 1 unit up and expanded vertically to graph g(x). D.f(x) = | x |; f(x) is translated 3 units left, 1 unit down and expanded horizontally to graph g(x). 5–Minute Check 3

  4. Find [f ○ g](x) and [g ○ f ](x) for f(x) = 2x – 4 andg(x) = x2. A.(2x – 4)x2; x2(2x – 4) B.4x2 – 16x + 16; 2x2 – 4 C.2x2 – 4; 4x2 – 16x + 16 D.4x2 – 4; 4x2 + 16 5–Minute Check 4

  5. Evaluate f(2x) if f(x) = x2 + 5x + 7. A.2x2 + 10x + 7 B.2x3 + 10x2 + 7 C.4x2 + 10x + 7 D.4x2 + 7x + 7 5–Minute Check 5

  6. Key Concept 1

  7. A. Graph and analyze . Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. Analyze Monomial Functions Evaluate the function for several x-values in its domain. Then use a smooth curve to connect each of these points to complete the graph. Example 1

  8. end behavior: Analyze Monomial Functions D = (–∞, ∞); R = [0, ∞); intercept: 0; continuity: continuous for all real numbers; decreasing: (–∞, 0); increasing: (0, ∞) Example 1

  9. Answer: D = (–∞, ∞); R = [0, ∞); intercept: 0; continuous for all real numbers; decreasing: (–∞, 0) , increasing: (0, ∞) Analyze Monomial Functions Example 1

  10. Analyze Monomial Functions B. Graph and analyze f(x) = –x5. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. Example 1

  11. end behavior: Analyze Monomial Functions D = (–∞, ∞); R = (–∞, ∞); intercept: 0; continuity: continuous for all real numbers; decreasing: (–∞, ∞) Example 1

  12. Answer: D = (–∞, ∞); R = (–∞, ∞);intercept: 0; continuous for all real numbers; decreasing: (–∞, ∞) Analyze Monomial Functions Example 1

  13. Describe where the graph of the function f(x) = 2x4 is increasing or decreasing. A. increasing:(–∞, ∞) B. decreasing: (–∞, 0) , increasing: (0, ∞) C. decreasing: (–∞, ∞) D. increasing: (–∞, 0), decreasing: (0, ∞) Example 1

  14. Functions with Negative Exponents A. Graph and analyze f(x) = 2x–4. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. Example 2

  15. end behavior: Functions with Negative Exponents intercept: none; continuity: infinite discontinuity at x = 0; increasing: (–∞, 0); decreasing: (0, ∞) Example 2

  16. Answer:D = (– ∞, 0)  (0, ∞); R = (0, ∞); no intercept ; infinite discontinuity at x = 0; increasing: (–∞, 0), decreasing: (0, ∞); Functions with Negative Exponents Example 2

  17. Functions with Negative Exponents B. Graph and analyze f(x) = 2x–3. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. Example 2

  18. end behavior: Functions with Negative Exponents D = (–∞, 0)  (0, ∞); R = (–∞, 0)  (0, ∞); intercept: none; continuity: infinite discontinuity at x = 0; decreasing: (–∞, 0) and (0, ∞) Example 2

  19. Answer:D = (–∞, 0)  (0, ∞); R = (–∞, 0)  (0, ∞); no intercept ;infinite discontinuity at x = 0; decreasing: (–∞, 0) and (0, ∞) Functions with Negative Exponents Example 2

  20. A. B. C. D. Describe the end behavior of the graph of f(x) = 3x–5. Example 2

  21. A. Graph and analyze . Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. Rational Exponents Example 3

  22. end behavior: Rational Exponents D = [0, ∞); R = [0, ∞); intercept: 0; continuity: continuous on [0, ∞); increasing: [0, ∞) Example 3

  23. Answer:D = [0, ∞); R = [0, ∞); intercept: 0; ;continuous on [0, ∞); increasing: [0, ∞) Rational Exponents Example 3

  24. B. Graph and analyze . Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. Rational Exponents Example 3

  25. end behavior: Rational Exponents D = (0, ∞); R = (0, ∞); intercept: none; continuity: continuous on (0, ∞); decreasing: (0, ∞) Example 3

  26. Answer:D = (0, ∞); R = (0, ∞); no intercept ; continuous on (0, ∞); decreasing: (0, ∞) Rational Exponents Example 3

  27. Describe the continuity of the function . A. continuous for all real numbers B. continuous on and C. continuous on (0, ∞] D. continuous on [0, ∞) Example 3

  28. Power Regression A. ANIMALS The following data represent the body length L in centimeters and the mass M in kilograms of several African Golden cats being studied by a scientist. Create a scatter plot of the data. Example 4

  29. Power Regression Answer: The scatter plot appears to resemble the square root function which is a power function. Example 4

  30. Power Regression B. ANIMALS The following data represent the body length L in centimeters and the mass M in kilograms of several African Golden cats being studied by a scientist. Determine a power function to model the data. Describe its end behavior Example 4

  31. Power Regression Using the PwrReg tool on a graphing calculator yields y = 0.018x1.538. The correlation coefficient r for the data, 0.875, suggests that a power regression accurately reflects the data. Answer:L = 0.018M1.538 Example 4

  32. Power Regression C. ANIMALS The following data represent the body length L in centimeters and the mass M in kilograms of several African Golden cats being studied by a scientist. Use the data to predict the mass of an African Golden cat with a length of 77 centimeters. Example 4

  33. Power Regression Use the CALC feature on the calculator to find f(77). The value of f(77) is about 14.1, so the mass of an African Golden cat with a length of 77 centimeters is about 14.1 kilograms. Answer:14.1 kg Example 4

  34. AIR The table shows the amount of air f(r) in cubic inches needed to fill a ball with a radius of r inches. Determine a power function to model the data. A. f(r) = 5.9r2.6 B. f(r) = 0.6r0.3 C. f(r) = 19.8(1.8)r D. f(r) = 5.2r2.9 Example 4

  35. Key Concept 5

  36. A. Graph and analyze . Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. Graph Radical Functions Example 5

  37. Graph Radical Functions D = [0, ∞); R = [0, ∞); intercept: 0; continuous on [0, ∞); increasing: [0, ∞) Example 5

  38. Answer:D = [0, ∞); R = [0, ∞); intercept: 0; ; continuous on [0, ∞); increasing: [0, ∞) Graph Radical Functions Example 5

  39. B. Graph and analyze . Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. Graph Radical Functions Example 5

  40. x-intercept: , y-intercept: about –0.6598; ; Graph Radical Functions D = (–∞, ∞); R = (–∞, ∞); continuous for all real numbers; increasing: (–∞, ∞) Example 5

  41. Answer:D = (–∞, ∞) ; R = (–∞, ∞) ; x-intercept: , y-intercept: about –0.6598; ; continuous for all real numbers; increasing: (–∞, ∞) Graph Radical Functions Example 5

  42. Find the intercepts of the graph of . A. x-intercept: , y-intercept: B. x-intercepts: , y-intercept: C. x-intercept: , y-intercept: D. x-intercepts: , y-intercept –4 Example 5

  43. A. Solve . Solve Radical Equations original equation Isolate the radical. Square each side to eliminate the radical. Subtract 28x and 29 from each side. Factor. Factor. x – 5 = 0 or x + 1 = 0 Zero Product Property x = 5 x = –1 Solve. Example 6

  44. Solve Radical Equations Answer:–1, 5 Check x = 5 x = –1 10 = 10  –2 = –2  A check of the solutions in the original equation confirms that the solutions are valid. Example 6

  45. B. Solve . Solve Radical Equations original equation Subtract 8 from each side. Raise each side to the third power. (The index is 3.) Take the square root of each side. x = 10 or –6 Add 2 to each side. A check of the solutions in the original equation confirms that the solutions are valid. Answer:10, –6 Example 6

  46. C. Solve . Solve Radical Equations original equation Square each side. Isolate the radical. Square each side. Distributive Property Combine like terms. (x – 8)(x – 24) = 0 Factor. Zero Product Property x – 8 = 0 or x – 24 = 0 Example 6

  47. Solve Radical Equations Solve. x = 8 x = 24 One solution checks and the other solution does not. Therefore, the solution is 8. Answer:8 Example 6

  48. Solve . A. 0, 5 B. 11, –11 C. 11 D. 0, 11 Example 6

  49. LESSON 2–1 Power and Radical Functions

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