Mid-Chapter Quiz: Lessons 2-1 through 2-3

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1 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 The function is a monomial with an even degree and a positive value for a. All values of x are included in the graph, so the function exists for all values of x, and D = (, ). All values of y are included in the graph, so the function exists for all values of y, and R = (, ). The only time the graph intersects the axes is when it goes through the origin, so the x- and y-intercepts are both 0. The y-values approach negative infinity as x approaches negative infinity, and positive infinity as x approaches positive infinity, so and. There are no breaks, holes, or gaps in the graph, so it is continuous for all real numbers. As you read the graph from left to right, it is going up from negative infinity to positive infinity, so the graph is increasing on (, ). esolutions Manual - Powered by Cognero Page 1

2 f (x) = 3x The function is a monomial with an even degree and a positive value for a. All values of x are included in the graph, so the function exists for all values of x, and D = (, ). The only values of y that are included in the graph are negative infinity through 0, so R = (, 0]. The only time the graph intersects the axes is when it goes through the origin, so the x- and y-intercepts are both 0. The y-values approach negative infinity as x approaches negative or positive infinity, so and. There are no breaks, holes, or gaps in the graph, so it is continuous for all real numbers. Since the power is negative, the function will be undefined at x = 0, and D = (, 0) (0, ). The only values of y that are included in the graph are greater than 0 through infinity, so R = (0, ). The graph never intersects either axis, so there are no x- and y-intercepts. The y-values approach zero as x approaches negative or positive infinity, so. The graph has an infinite discontinuity at x = 0. and As you read the graph from left to right, it is going up from negative infinity to 0, and then going up again from 0 to positive infinity, so the graph is increasing on (, 0) (0, ). As you read the graph from left to right, it is going up from negative infinity to 0, and then going down from 0 to positive infinity, so the graph is increasing on (, 0) and decreasing on (0, ). esolutions Manual - Powered by Cognero Page 2

3 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. The function is a power function with a rational exponent and a positive value for a. All values of x are included in the graph, so the function exists for all values of x, and D = (, ). The only values of y that are included in the graph are 0 through infinity, so R = [0, ). The only time the graph intersects the axes is when it goes through the origin, so the x- and y-intercepts are both 0. The y-values approach positive infinity as x approaches negative or positive infinity, so and. 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 value function from the CALC menu on the graphing calculator. Let x = 7.6. There are no breaks, holes, or gaps in the graph, so it is continuous for all real numbers. As you read the graph from left to right, it is going down from negative infinity to 0, and then going up from 0 to positive infinity, so the graph is decreasing on (, 0) and increasing on (0, ). The area under the branches of a fir tree that is 7.6 meters high is about square meters. esolutions Manual - Powered by Cognero Page 3

4 Solve each equation. 6. = = Since the each side of the equation was raised to a power, check the solution in the original equation. 7. The solution is 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: 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 The solution is 2. x = 1 The solutions are 1 and 3. esolutions Manual - Powered by Cognero Page 4

5 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. Since the each side of the equation was raised to a power, check the solution in the original equation. The zeros are 1, 0, and. 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. 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. Because ± are not real zeros, f has two distinct real zeros, 1 and 1. So, the zeros are 2 and 13. esolutions Manual - Powered by Cognero Page 5

6 13. MULTIPLE CHOICE Which of the following de behavior of a polynomial of odd degree? A B C D A polynomial of odd degree with either have an end Crystal will use the following January. Does this answer make sense? Explain your reasoning. 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. The correct answer is D. 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 x + 7 The degree is 4 and the leading coefficient is 7. Because the degree is even and the leading coefficient is negative, f(x) = 2.707x x x 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. Since the following January is 12 months from the first January, let x = f (x) = 5x 5 + 4x x 2 8 The degree is 5 and the leading coefficient is 5. Because the degree is odd and the leading coefficient is negative, 16. ENERGY Crystal s electricity consumption measured in kilowatt hours (kwh) for the past 12 months is shown below. The amount of kilowatt hours Crystal will use the following January is about This answer does not make sense because it is not possible to consume negative kwh in a month. 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 esolutions Manual - Powered by Cognero Page 6