1. Graph and analyze power functions.
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.
Then/Now

2. power function monomial function radical function extraneous solution
Vocabulary

3. Key Concept 1

4. 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

5. 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

6. 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

7. 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

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

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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

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

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

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

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

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

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

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

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

24. 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

25. 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

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

27. 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

28. Power Regression
Using the PwrReg tool on a graphing calculator yields y = 0.018x 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

29. 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

30. 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: kg
Example 4

31. 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

32. Key Concept 5

33. 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

34. 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

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

36. 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

37. 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

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

39. 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

40. 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

41. 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

42. 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

43. 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

44. 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

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

46. End of the Lesson