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Lesson Menu 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 5–Minute Check 1 Determine whether the relation y 2 – 4x = 3 represents y as a function of x. A.yes B.no

Over Chapter 1 5–Minute Check 2 Describe the end behavior of f (x) = 4x 4 + 2x – 8. A. B. C. D.

Over Chapter 1 5–Minute 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).

Over Chapter 1 5–Minute 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

Over Chapter 1 5–Minute Check 5 Evaluate f (2x) if f (x) = x 2 + 5x + 7. A.2x x + 7 B.2x x C.4x x + 7 D.4x 2 + 7x + 7

Then/Now 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.

Vocabulary power function monomial function radical function extraneous solution

Key Concept 1

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.

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

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

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.

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

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

Example 1 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, ∞)

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.

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

Example 2 Functions with Negative Exponents Answer: D = (– ∞, 0)  (0, ∞); R = (0, ∞); ; infinite discontinuity at x = 0; increasing: (–∞, 0), decreasing: (0, ∞);

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.

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, ∞);

Example 2 Functions with Negative Exponents Answer: D = (–∞, 0)  (0, ∞); R = (–∞, 0)  (0, ∞); ; infinite discontinuity at x = 0; decreasing: (–∞, 0) and (0, ∞)

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

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

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

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

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

Example 3 Rational Exponents intercept: none; continuity: infinite discontinuity at x = 0; end behavior: increasing: (–∞, 0), decreasing: (0, ∞) D = (–∞, 0)  (0, ∞); R = (0, ∞);

Example 3 Rational Exponents Answer: D = (–∞, 0)  (0, ∞); R = (0, ∞); ; infinite discontinuity at x = 0; increasing: (–∞, 0), decreasing: (0, ∞)

Example 3 A.continuous for all real numbers B.continuous on and C.continuous on D.continuous on Describe the continuity of the function.

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.

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

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.

Example 4 Answer:y = 0.018x Power Regression Using the PwrReg tool on a graphing calculator yields f (x) = 0.018x The correlation coefficient r for the data, 0.875, suggests that a power regression may accurately reflect the data.

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.

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.

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

Key Concept 5

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.

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

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

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.

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

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

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

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.

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

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

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

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

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

End of the Lesson