Five-Minute Check (over Chapter 1) Then/Now New Vocabulary Key Concept: Monomial Functions Example 1: Analyze Monomial Functions Example 2: Functions with Negative Exponents Example 3: Rational Exponents Example 4: Power Regression Key Concept: Radical Functions Example 5: Graph Radical Functions Example 6: Solve Radical Equations Over Chapter 1 Determine whether the relation y 2 – 4x = 3 represents y as a function of x. A. yes B. no Over Chapter 1 Describe the end behavior of f(x) = 4x 4 + 2x – 8. A. B. C. D. Over Chapter 1 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). Over Chapter 1 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 Over Chapter 1 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 You analyzed parent functions and their families of graphs. (Lesson 1-5) • Graph and analyze power functions. • Graph and analyze radical functions, and solve radical equations. • power function • monomial function • radical function • extraneous solution Analyze Monomial Functions A. Graph and analyze . Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. 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. Analyze Monomial Functions D = (–∞, ∞); R = [0, ∞); intercept: 0; end behavior: continuity: continuous for all real numbers; decreasing: (–∞, 0); increasing: (0, ∞) Analyze Monomial Functions Answer: D = (–∞, ∞); R = [0, ∞); intercept: 0; continuous for all real numbers; decreasing: (–∞, 0) , increasing: (0, ∞) 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. Analyze Monomial Functions D = (–∞, ∞); R = (–∞, ∞); intercept: 0; end behavior: continuity: continuous for all real numbers; decreasing: (–∞, ∞) Analyze Monomial Functions Answer: D = (–∞, ∞); R = (–∞, ∞); intercept: 0; continuous for all real numbers; decreasing: (–∞, 0) Describe where the graph of the function f(x) = 2x 4 is increasing or decreasing. A. increasing: (–∞, ∞) B. increasing: (–∞, 0) , decreasing: (0, ∞) C. decreasing: (–∞, ∞) D. increasing: (–∞, 0), decreasing: (0, ∞) 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. Functions with Negative Exponents intercept: none; end behavior: continuity: infinite discontinuity at x = 0; increasing: (–∞, 0); decreasing: (0, ∞) Functions with Negative Exponents Answer: D = (– ∞, 0) (0, ∞); R = (0, ∞); ; infinite discontinuity at x = 0; increasing: (–∞, 0), decreasing: (0, ∞); 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. Functions with Negative Exponents D = (–∞, 0) (0, ∞); R = (–∞, 0) (0, ∞); intercept: none; end behavior: continuity: infinite discontinuity at x = 0; decreasing: (–∞, 0) and (0, ∞) Functions with Negative Exponents Answer: D = (–∞, 0) (0, ∞); R = (–∞, 0) (0, ∞); ; infinite discontinuity at x = 0; decreasing: (–∞, 0) and (0, ∞) Describe the end behavior of the graph of f(x) = 3x –5. A. B. C. D. Rational Exponents A. Graph and analyze . Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. Rational Exponents D = [0, ∞); R = [0, ∞); intercept: 0; end behavior: continuity: continuous on [0, ∞); increasing: (0, ∞) Rational Exponents Answer: D = [0, ∞); R = [0, ∞); intercept: 0; ; continuous on (0, ∞); increasing: (0, ∞) Rational Exponents B. Graph and analyze . Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. Rational Exponents D = (–∞, 0) (0, ∞); R = (0, ∞); intercept: none; end behavior: continuity: infinite discontinuity at x = 0; increasing: (–∞, 0), decreasing: (0, ∞) Rational Exponents Answer: D = (–∞, 0) (0, ∞); R = (0, ∞); ; infinite discontinuity at x = 0; increasing: (–∞, 0), decreasing: (0, ∞) Describe the continuity of the function A. continuous for all real numbers B. continuous on C. continuous on D. continuous on and . 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. Power Regression Answer: The scatter plot appears to resemble the square root function which is a power function. 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. Power Regression Using the PwrReg tool on a graphing calculator yields f(x) = 0.018x1.538. The correlation coefficient r for the data, 0.875, suggests that a power regression may accurately reflect the data. Answer: y = 0.018x 1.538 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. 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 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 Graph Radical Functions A. Graph and analyze . Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. Graph Radical Functions D = [0, ∞); R = [0, ∞); intercept: 0; continuous on [0, ∞); increasing: (0, ∞) Graph Radical Functions Answer: D = [0, ∞); R = [0, ∞); intercept: 0; ; continuous on [0, ∞); increasing: (0, ∞) Graph Radical Functions B. Graph and analyze . Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. Graph Radical Functions D = (–∞, ∞); R = (–∞, ∞); x-intercept: , y-intercept: about –0.6598; ; continuous for all real numbers; increasing: (–∞, ∞) Graph Radical Functions Answer: D = (–∞, ∞) ; R = (–∞, ∞) ; x-intercept: y-intercept: about –0.6598; ; continuous for all real numbers; increasing: (–∞, ∞) , Find the intercepts of the graph of A. x-intercept: B. x-intercepts: C. x-intercept: D. x-intercepts: , y-intercept: , y-intercept: , y-intercept: , y-intercept –4 . Solve Radical Equations A. Solve . 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 x =5 x = –1 Zero Product Property Solve. Solve Radical Equations Answer: –1, 5 Check x = –1 x=5 10 = 10 –2 = –2 A check of the solutions in the original equation confirms that the solutions are valid. 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. 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 Solve Radical Equations C. Solve . original equation Square each side. Isolate the radical. Square each side. (x – 8)(x – 24) = 0 x – 8 = 0 or x – 24 = 0 Distributive Property Combine like terms. Factor. Zero Product Property Solve Radical Equations x= 8 x = 24 Solve. One solution checks and the other solution does not. Therefore, the solution is 8. Answer: 8 Solve A. 0, 5 B. 11, –11 C. 11 D. 0, 11 .