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Power and Radical Functions LESSON 21 Slide 2 Over Chapter 1 5Minute Check 2 Describe the end behavior of f (x) = 4x 4 + 2x 8. A. B. C. D. Slide 3 Over Chapter 1 5Minute Check 3 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). C > 1 expands it 0 < C < 1 compresses it Slide 4 Over Chapter 1 5Minute Check 4 Find [f g](x) and [g f ](x) for f (x) = 2x 4 and g (x) = x 2. A.(2x 4)x 2 ; x 2 (2x 4) B.4x 2 16x + 16; 2x 2 4 C.2x 2 4; 4x 2 16x + 16 D.4x 2 4; 4x 2 + 16 Slide 5 Over Chapter 1 5Minute Check 5 Evaluate f (2x) if f (x) = x 2 + 5x + 7. A.2x 2 + 10x + 7 B.2x 3 + 10x 2 + 7 C.4x 2 + 10x + 7 D.4x 2 + 7x + 7 Slide 6 Key Concept 1 Slide 7 Example 1 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. A. Graph and analyze. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. Slide 8 Example 1 Analyze Monomial Functions intercept: 0; continuity: continuous for all real numbers; decreasing: (, 0); increasing: (0, ) end behavior: D = (, ); R = [0, ); Slide 9 Example 1 Analyze Monomial Functions Answer: D = ( , ); R = [0, ); intercept: 0; continuous for all real numbers; decreasing: (, 0), increasing: (0, ) Slide 10 Example 1 Analyze Monomial Functions B. Graph and analyze f (x) = x 5. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. Slide 11 Example 1 Analyze Monomial Functions intercept: 0; end behavior: continuity: continuous for all real numbers; decreasing: (, ) D = (, ); R = (, ); Slide 12 Example 1 Analyze Monomial Functions Answer: D = (, ); R = (, ); intercept: 0; continuous for all real numbers; decreasing: (, ) Slide 13 Example 1 Describe where the graph of the function f (x) = 2x 4 is increasing or decreasing. A.increasing: (, ) B.decreasing: (, 0), increasing: (0, ) C.decreasing: (, ) D.increasing: (, 0), decreasing: (0, ) Slide 14 Example 2 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. Slide 15 Example 2 Functions with Negative Exponents intercept: none; continuity: infinite discontinuity at x = 0; end behavior: increasing: (, 0); decreasing: (0, ) Slide 16 Example 2 Functions with Negative Exponents Answer: D = ( , 0) (0, ); R = (0, ); no intercept ; infinite discontinuity at x = 0; increasing: (, 0), decreasing: (0, ); Slide 17 Example 2 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. Slide 18 Example 2 Functions with Negative Exponents intercept: none; continuity: infinite discontinuity at x = 0; end behavior: decreasing: (, 0) and (0, ) D = (, 0) (0, ); R = (, 0) (0, ); Slide 19 Example 2 Functions with Negative Exponents Answer: D = (, 0) (0, ); R = (, 0) (0, ); no intercept ; infinite discontinuity at x = 0; decreasing: (, 0) and (0, ) Slide 20 Example 2 Describe the end behavior of the graph of f (x) = 3x 5. A. B. C. D. Slide 21 Example 3 Rational Exponents A. Graph and analyze. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. Slide 22 Example 3 Rational Exponents intercept: 0; end behavior: increasing: [0, ) continuity: continuous on [0, ); D = [0, ); R = [0, ); Slide 23 Example 3 Rational Exponents Answer: D = [0, ); R = [0, ); intercept: 0; ; continuous on [0, ); increasing: [0, ) Slide 24 Example 3 Rational Exponents B. Graph and analyze. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. Slide 25 Example 3 Rational Exponents intercept: none; continuity: continuous on (0, ); end behavior: decreasing: (0, ) D = (0, ); R = (0, ); Slide 26 Example 3 Rational Exponents Answer: D = (0, ); R = (0, ); no intercept ; continuous on (0, ); decreasing: (0, ) Slide 27 Example 3 A.continuous for all real numbers B.continuous on and C.continuous on (0, ] D.continuous on [0, ) Describe the continuity of the function. Slide 28 Example 4 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. Slide 29 Example 4 Answer: Power Regression The scatter plot appears to resemble the square root function which is a power function. Slide 30 Example 4 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 Slide 31 Example 4 Answer:L = 0.018M 1.538 Power Regression Using the PwrReg tool on a graphing calculator yields y = 0.018x 1.538. The correlation coefficient r for the data, 0.875, suggests that a power regression accurately reflects the data. Slide 32 Example 4 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. Slide 33 Example 4 Answer: 14.1 kg 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. Slide 34 Example 4 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) = 5.9r 2.6 B.f (r) = 0.6r 0.3 C.f (r) = 19.8(1.8) r D.f (r) = 5.2r 2.9 Slide 35 Key Concept 5 Slide 36 Example 5 Graph Radical Functions A. Graph and analyze. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. Slide 37 Example 5 Graph Radical Functions intercept: 0; increasing: [0, ) continuous on [0, ); D = [0, ); R = [0, ); Slide 38 Example 5 Graph Radical Functions Answer:D = [0, ); R = [0, ); intercept: 0; ; continuous on [0, ); increasing: [0, ) Slide 39 Example 5 Graph Radical Functions B. Graph and analyze. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. Slide 40 Example 5 Graph Radical Functions continuous for all real numbers; ; x-intercept:, y-intercept: about 0.6598; increasing: (, ) D = (, ); R = (, ); Slide 41 Example 5 Graph Radical Functions Answer:D = (, ) ; R = (, ) ; x-intercept:, y-intercept: about 0.6598; ; continuous for all real numbers; increasing: (, ) Slide 42 Example 5 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 Slide 43 Example 6 original equation Solve Radical Equations A. Solve. Isolate the radical. Square each side to eliminate the radical. Subtract 28x and 29 from each side. Factor. x 5= 0 or x + 1= 0 Zero Product Property x = 5 x= 1Solve. Slide 44 Example 6 Solve Radical Equations Check x = 5 10 = 10 x = 1 2 = 2 A check of the solutions in the original equation confirms that the solutions are valid. Answer: 1, 5 Slide 45 Example 6 Solve Radical Equations B. Solve. 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. Add 2 to each side.x = 10 or 6 A check of the solutions in the original equation confirms that the solutions are valid. Answer: 10, 6 Slide 46 Example 6 Solve Radical Equations C. Solve. original equation Square each side. Isolate the radical. Distributive Property Combine like terms. Square each side. Factor. Zero Product Property (x 8)(x 24) = 0 x 8 = 0 or x 24 = 0 Slide 47 Example 6 Solve Radical Equations Solve. x= 8 x= 24 One solution checks and the other solution does not. Therefore, the solution is 8. Answer: 8 Slide 48 Example 6 A.0, 5 B.11, 11 C.11 D.0, 11 Solve. Slide 49 Power and Radical Functions LESSON 21