Power and Radical Functions LESSON 2–1 Power and Radical Functions

Describe the end behavior of f (x) = 4x 4 + 2x – 8. 5–Minute Check 2

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

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 5–Minute Check 4

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 5–Minute Check 5

Key Concept 1

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. Example 1

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

Answer: D = (–∞, ∞); R = [0, ∞); intercept: 0; 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

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

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

B. decreasing: (–∞, 0) , increasing: (0, ∞) C. decreasing: (–∞, ∞) 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, ∞) Example 1

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

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

Answer: D = (– ∞, 0)  (0, ∞); R = (0, ∞); no intercept ; Functions with Negative Exponents Answer: D = (– ∞, 0)  (0, ∞); R = (0, ∞); no intercept ; 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

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

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

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

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

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

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

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

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

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

Power Regression Answer: 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. Describe its end behavior Example 4

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 = 0.018M 1.538 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

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

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 Example 4

Key Concept 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

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

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

Square each side to eliminate the radical. 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 Zero Product Property x = 5 x = –1 Solve. Example 6

Answer: –1, 5 Check x = 5 x = –1 10 = 10  –2 = –2  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

Subtract 8 from each side. 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 Example 6

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

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

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

Power and Radical Functions LESSON 2–1 Power and Radical Functions