Graph and analyze each function. Describe its domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. 1. f (x) = 2x 3 Evaluate the function for several x-values in its domain. x 2 1.5 1 0 1 1.5 2 f (x) 16 6.75 2 0 2 6.75 16 Use these points to construct a graph. The function is a monomial with an odd degree and a positive value for a. D = (, ), R = (, ); intercept: 0; continuous for all real numbers; increasing: (, ) 2. Evaluate the function for several x-values in its domain. x 2 1.5 1 0 1 1.5 2 f (x) 10.7 3.4 0.7 0 0.7 3.4 10.7 Use these points to construct a graph. The function is a monomial with an even degree and a negative value for a. D = (, ), R = (, 0]; intercept: 0; continuous for all real numbers; increasing: (, 0); decreasing: (0, ) esolutions Manual - Powered by Cognero Page 1
3. f (x) = 3x 8 Evaluate the function for several x-values in its domain. x 2 1 0.5 0 0.5 1 2 f (x) 0.01 3 768 768 3 0.01 Use these points to construct a graph. Since the power is negative, the function will be undefined at x = 0. D = (, 0) (0, ), R = (0, ); no intercepts; infinite discontinuity at x = 0; increasing: (, 0); decreasing: (0, ) 4. Evaluate the function for several x-values in its domain. x 4 2 1 0 1 2 4 f (x) 6.97 5.28 4 0 4 5.28 6.97 Use these points to construct a graph. D = (, ), R = [0, ); intercept: 0; continuous for all real numbers; decreasing: (, 0); increasing: (0, ) esolutions Manual - Powered by Cognero Page 2
5. TREES The heights of several fir trees and the areas under their branches are shown in the table. a. Create a scatter plot of the data. b. Determine a power function to model the data. c. Predict the area under the branches of a fir tree that is 7.6 meters high. a. Enter the data into a graphing calculator and create a scatter plot. b. Use the power regression function on the graphing calculator to find values for a and n. f(x) = 1.37x 2.3 c. Graph the regression equation using a graphing calculator. To predict the area under the branches of a fir tree that is 7.6 meters high, use the CALC function on the graphing calculator. Let x = 7.6. The area under the branches of a fir tree that is 7.6 meters high is about 149.26 square meters. esolutions Manual - Powered by Cognero Page 3
Solve each equation. 6. = 13 Since the each side of the equation was raised to a power, check the solution in the original equation. The solution is 32.4. 7. + 1 = x x = 3 or x = 1. Since the each side of the equation was raised to a power, check the solutions in the original equation. x = 3 x = 1 The solutions are 1 and 3. esolutions Manual - Powered by Cognero Page 4
8. + 1 = x = 5 or x = 2. Since the each side of the equation was raised to a power, check the solutions in the original equation. x = 5 x = 2 The solution is 2. esolutions Manual - Powered by Cognero Page 5
9. Since the each side of the equation was raised to a power, check the solution in the original equation. The solution is 13. State the number of possible real zeros and turning points of each function. Then find all of the real zeros by factoring. 10. f (x) = x 2 11x 26 The degree of f (x) is 2, so it will have at most two real zeros and one turning point. So, the zeros are 2 and 13. 11. f (x) = 3x 5 + 2x 4 x 3 The degree of f (x) is 5, so it will have at most five real zeros and four turning points. x = 1, 0, or So, the zeros are 1,0, and. esolutions Manual - Powered by Cognero Page 6
12. f (x) = x 4 + 9x 2 10 The degree of f (x) is 4, so it will have at most four real zeros and three turning points. Let u = x 2. x = 1, 1 or Because ± are not real zeros, f has two distinct real zeros, 1 and 1. 13. MULTIPLE CHOICE Which of the following describes the possible end behavior of a polynomial of odd degree? A B C D A polynomial of odd degree with either have an end behavior of Describe the end behavior of the graph of each polynomial function using limits. Explain your reasoning using the leading term test. 14. f (x) = 7x 4 3x 3 8x 2 +23x + 7 The degree is 4 and the leading coefficient is 7. Because the degree is even and the leading coefficient is negative, 15. f (x) = 5x 5 + 4x 4 + 12x 2 8 The degree is 5 and the leading coefficient is 5. Because the degree is odd and the leading coefficient is negative, esolutions Manual - Powered by Cognero Page 7
16. ENERGY Crystal s electricity consumption measured in kilowatt hours (kwh) for the past 12 months is shown below. a. Determine a model for the number of kilowatt hours Crystal used as a function of the number of months since January. b. Use the model to predict how many kilowatt hours Crystal will use the following January. Does this answer make sense? Explain your reasoning. Table being fixed a. Sample answer: Since the data shows two possible turning points, a third-degree polynomial may be the best model to represent the data. Use the cubic regression function on the graphing calculator. Let x = 0 represent January. f(x) = 2.707x 3 + 41.392x 2 141.452x + 238.176 b. Sample answer: Graph the regression equation using a graphing calculator. To predict how many kilowatt hours Crystal will use the following January, use the CALC function on the graphing calculator. Since the following January is 12 months since the first January, let x = 12. The amount of kilowatt hours Crystal will use the following January is about 177.273. This answer does not make sense because it is not possible to consume negative kwh in a month. esolutions Manual - Powered by Cognero Page 8
Divide using synthetic division. 17. (5x 3 7x 2 + 8x 13) (x 1) Because x 1, c = 1. Set up the synthetic division as follows. Then follow the synthetic division procedure. The quotient is. 18. (x 4 x 3 9x + 18) (x 2) Because x 2, c = 2. Set up the synthetic division as follows. Then follow the synthetic division procedure. The quotient is x 3 + x 2 + 2x 5 +. 19. (2x 3 11x 2 + 9x 6) (2x 1) Rewrite the division expression so that the divisor is of the form x c. Because c =. Set up the synthetic division as follows, using a zero placeholder for the missing x-term in the dividend. Then follow the synthetic division procedure. The remainder can be written as. So, the quotient is. esolutions Manual - Powered by Cognero Page 9
Determine each f (c) using synthetic substitution. 20. f (x) = 9x 5 + 4x 4 3x 3 + 18x 2 16x + 8; c = 2 The remainder is 376. Therefore, f (2) = 376. 21. f (x) = 6x 6 3x 5 + 8x 4 + 12x 2 6x + 4; c = 3 The remainder is 5881. Therefore, f ( 3) = 5881. 22. f (x) = 2x 6 + 8x 5 12x 4 + 9x 3 8x 2 + 6x 3; c = 2 The remainder is 695. Therefore, f ( 2) = 695. Use the Factor Theorem to determine if the binomials given are factors of f (x). Use the binomials that are factors to write a factored form of f (x). 23. f (x) = x 3 + 2x 2 25x 50; (x + 5) Use synthetic division to test (x + 5). Because the remainder when f (x) is divided by (x + 5) is 0, (x + 5) is a factor. We can use the quotient of f (x) (x + 5) to write a factored form of f (x) as f (x) = (x + 5)(x 2 3x 10). Factoring the quadratic expression yields f (x) = (x + 5)(x 5)(x + 2). esolutions Manual - Powered by Cognero Page 10
24. f (x) = x 4 6x 3 + 7x 2 + 6x 8; (x 1), (x 2) Use synthetic division to test each factor, (x 1) and (x 2). Because the remainder when f (x) is divided by (x 1) is 0, (x 1) is a factor. Test the second factor, (x 2), with the depressed polynomial x 3 5x 2 + 2x + 8. Because the remainder when the depressed polynomial is divided by (x 2) is 0, (x 2) is a factor of f (x). Because (x 1) and (x 2) are factors of f (x), we can use the final quotient to write a factored form of f (x) as f (x) = (x 1)(x 2)(x 2 3x 4). Factoring the quadratic expression yields f (x) = (x 1)(x 2)(x 4)(x + 1). 25. MULTIPLE CHOICE Find the remainder when f (x) = x3 4x + 5 is divided by x + 3. F 10 G 8 H 20 J 26 Because x + 3, c = 3. Set up the synthetic division as follows. Then follow the synthetic division procedure. The quotient is 10. The correct answer is F. esolutions Manual - Powered by Cognero Page 11