Algebra 2 Spring 2015 Test 1

Transcription

1 Name: Class: Date: Algebra 2 Spring 2015 Test 1 Multiple Choice (23 questions) Identify the choice that best completes the statement or answers the question on your scantron sheet. Do any scratch work neatly to the problems for partial credit. 1. Which is the graph of y = 2(x 2) 2 4? a. c. b. d. Factor the expression. 2. x 2 6x + 8 a. (x + 4)(x + 2) c. (x 4)(x + 2) b. (x 2)(x 4) d. (x 2)(x + 4) 1

2 Name: Solve the equation by finding square roots. 3. 3x 2 = a. 7 c., 3 3 b. 7, 7 d. 7, Simplify 175 using the imaginary number i. a. i 175 b. 5i 7 c. 5 7 d. 5 7 Write the number in the form a + bi a i c i b i 4 d i a. 6 + i 48 c. 6 4i 3 b. 6 4i 3 d i 3 Simplify the expression. 7. ( 1 + 6i) + ( 4 + 2i) a. 5 8i c i b. 5 2i d. 3i 8. (2 5i) (3 + 4i) a i c. 1 9i b. 5 i d. 10i 2

3 Name: 9. (2 + 5i)( 1 + 5i) a i c i b i d i Solve the equation. 10. x x + 81 = 25 a. 14, 4 c. 14, 14 b. 4, 14 d. 4, Find the missing value to complete the square. x 2 + 2x + a. 2 b. 1 c. 4 d. 8 Solve the quadratic equation by completing the square. 12. x x + 14 = 0 a. 10 ± 6 c. 5 ± 6 b. 100 ± 11 d. 5 ± 11 3

4 Name: 13. x x + 35 = 0 a. 10 ± 15 c. 100 ± i 10 b. 5 ± i 15 d. 5 ± i 10 Rewrite the equation in vertex form. 14. y = x x + 16 a. y = (x + 5) c. y = (x + 10) b. y = (x + 10) 2 9 d. y = (x + 5) 2 9 Use the Quadratic Formula to solve the equation x 2 x + 3 = 0 a. 1 8 ± 47 8 b. 8 ± i 94 8 c. d. 1 8 ± i ± i

5 Name: Solve the equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. 16. x 2 + 5x + 4 = 0 a. c. b. The solution set is { 1, 4}. d. The solution set is { 2.5, 2.25}. The solution set is { 4, 1}. The solution set is { 1, 4}. Solve the equation by completing the square. 17. x 2 + 2x 3 = 0 a. { 3, 1} c. { 6, 1} b. { 6, 2} d. { 1, 3} 5

6 Name: 18. 2x 2 + 2x = 0 a. { 2, 0} c. {} 0 b. { 0, 1} d. { 1, 0} Find the exact solution of the following quadratic equation by using the Quadratic Formula. 19. x 2 8x = 20 a. { 10, 2} c. { 4, 20} b. { 20, 28} d. { 2, 10} Find the value of the discriminant. Then describe the number and type of roots for the equation. 20. x 2 14x + 2 = 0 a. The discriminant is 196. Because the discriminant is greater than 0 and is a perfect square, the two roots are real and rational. b. The discriminant is 204. Because the discriminant is less than 0, the two roots are complex. c. The discriminant is 204. Because the discriminant is greater than 0 and is not a perfect square, the two roots are real and irrational. d. The discriminant is 188. Because the discriminant is less than 0, the two roots are complex. 6

7 Name: 21. x 2 + x + 7 = 0 a. The discriminant is 29. Because the discriminant is less than 0, the two roots are complex. b. The discriminant is 1. Because the discriminant is greater than 0 and is a perfect square, the two roots are real and rational. c. The discriminant is 27. Because the discriminant is less than 0, the two roots are complex. d. The discriminant is 27. Because the discriminant is greater than 0 and is a perfect square, the two roots are real and rational. Write the following quadratic function in the vertex form. Then, identify the axis of symmetry. 22. y = x 2 + 4x 6 a. The vertex form of the function is y = ( x + 2) The equation of the axis of symmetry is x = 2. b. The vertex form of the function is y = ( x 2) The equation of the axis of symmetry is x = 2. c. The vertex form of the function is y = ( x + 2) The equation of the axis of symmetry is x = 10. d. The vertex form of the function is y = ( x + 2) The equation of the axis of symmetry is x = 10. Find the coordinates of the vertex of the quadratic function. 23. y = 8x 2 3 a. Ê Ë Á0, 3ˆ c. Ê Ë Á0, 3ˆ b. Ê Ë Á3, 0ˆ d. Ê Ë Á 3, 0ˆ 7

8 Name: Free Response (4 questions) 24. Graph y = (x 7) Determine the type and number of solutions of 4x 2 3x + 7 = 0. Number of solutions: Type of solutions (circle one): real, rational real, irrational complex 8

9 Name: 26. Consider the quadratic function fx ()= x 2 + 4x 5. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function. 27. Examine the pair of equations and write the similarities and differences in their graphs in no more than four brief and complete sentences. a. y = 1 3 x 2, y = 1 3 x Extra Credit (4 points maximum) Show that 2x 2 12x + 23 is equal to 2(x 3) Then use this to explain how you know that 5 is the minimum value of the function. 9

10 Algebra 2 Spring 2015 Test 1 Answer Section MULTIPLE CHOICE 1. ANS: A PTS: 1 DIF: L2 REF: 5-3 Translating Parabolas OBJ: Using Vertex Form NAT: NAEP G2c CAT5.LV21/22.54 CAT5.LV21/22.56 IT.LV17/18.AM S9.TSK3.GM S9.TSK3.PRA S10.TSK3.GM S10.TSK3.PRA TV.LV21/22.14 TV.LV21/22.16 TV.LVALG.56 TV.LVALG.57 STA: CA A2 9.0 TOP: 5-3 Example 1 KEY: graphing translation 2. ANS: B PTS: 1 DIF: L2 REF: 5-4 Factoring Quadratic Expressions OBJ: Finding Common and Binomial Factors NAT: NAEP A3b NAEP A3c CAT5.LV21/22.50 CAT5.LV21/22.52 IT.LV17/18.AM IT.LV17/18.CP S9.TSK3.PRA S10.TSK3.PRA TV.LV21/22.11 TV.LV21/22.52 TV.LVALG.53 TV.LVALG.57 STA: CA A2 4.0 TOP: 5-4 Example 3 KEY: factor a quadratic expression quadratic expression 3. ANS: B PTS: 1 DIF: L2 REF: 5-5 Quadratic Equations OBJ: Solving by Factoring and Finding Square Roots NAT: NAEP A4a CAT5.LV21/22.50 CAT5.LV21/22.52 CAT5.LV21/22.56 IT.LV17/18.AM IT.LV17/18.CP S9.TSK3.GM S9.TSK3.PRA S10.TSK3.GM S10.TSK3.PRA TV.LV21/22.12 TV.LV21/22.16 TV.LV21/22.17 TV.LVALG.56 TV.LVALG.57 STA: CA A2 8.0 CA A TOP: 5-5 Example 2 KEY: square root 4. ANS: B PTS: 1 DIF: L2 REF: 5-6 Complex Numbers OBJ: Identifying Complex Numbers NAT: NAEP A2c CAT5.LV21/22.49 CAT5.LV21/22.50 IT.LV17/18.AM IT.LV17/18.CP S9.TSK3.NS S9.TSK3.PRA S10.TSK3.NS S10.TSK3.PRA TV.LV21/22.10 TV.LV21/22.11 TV.LV21/22.12 TV.LVALG.53 STA: CA A2 5.0 CA A2 6.0 TOP: 5-6 Example 1 KEY: i imaginary number 5. ANS: C PTS: 1 DIF: L2 REF: 5-6 Complex Numbers OBJ: Identifying Complex Numbers NAT: NAEP A2c CAT5.LV21/22.49 CAT5.LV21/22.50 IT.LV17/18.AM IT.LV17/18.CP S9.TSK3.NS S9.TSK3.PRA S10.TSK3.NS S10.TSK3.PRA TV.LV21/22.10 TV.LV21/22.11 TV.LV21/22.12 TV.LVALG.53 STA: CA A2 5.0 CA A2 6.0 TOP: 5-6 Example 2 KEY: i imaginary number complex number 6. ANS: B PTS: 1 DIF: L2 REF: 5-6 Complex Numbers OBJ: Identifying Complex Numbers NAT: NAEP A2c CAT5.LV21/22.49 CAT5.LV21/22.50 IT.LV17/18.AM IT.LV17/18.CP S9.TSK3.NS S9.TSK3.PRA S10.TSK3.NS S10.TSK3.PRA TV.LV21/22.10 TV.LV21/22.11 TV.LV21/22.12 TV.LVALG.53 STA: CA A2 5.0 CA A2 6.0 TOP: 5-6 Example 2 KEY: i imaginary number complex number 7. ANS: C PTS: 1 DIF: L2 REF: 5-6 Complex Numbers OBJ: Operations With Complex Numbers NAT: NAEP A2c CAT5.LV21/22.49 CAT5.LV21/22.50 IT.LV17/18.AM IT.LV17/18.CP S9.TSK3.NS S9.TSK3.PRA S10.TSK3.NS S10.TSK3.PRA TV.LV21/22.10 TV.LV21/22.11 TV.LV21/22.12 TV.LVALG.53 STA: CA A2 5.0 CA A2 6.0 TOP: 5-6 Example 5 KEY: simplifying a complex number complex number 1

11 8. ANS: C PTS: 1 DIF: L2 REF: 5-6 Complex Numbers OBJ: Operations With Complex Numbers NAT: NAEP A2c CAT5.LV21/22.49 CAT5.LV21/22.50 IT.LV17/18.AM IT.LV17/18.CP S9.TSK3.NS S9.TSK3.PRA S10.TSK3.NS S10.TSK3.PRA TV.LV21/22.10 TV.LV21/22.11 TV.LV21/22.12 TV.LVALG.53 STA: CA A2 5.0 CA A2 6.0 TOP: 5-6 Example 5 KEY: simplifying a complex number complex number 9. ANS: A PTS: 1 DIF: L2 REF: 5-6 Complex Numbers OBJ: Operations With Complex Numbers NAT: NAEP A2c CAT5.LV21/22.49 CAT5.LV21/22.50 IT.LV17/18.AM IT.LV17/18.CP S9.TSK3.NS S9.TSK3.PRA S10.TSK3.NS S10.TSK3.PRA TV.LV21/22.10 TV.LV21/22.11 TV.LV21/22.12 TV.LVALG.53 STA: CA A2 5.0 CA A2 6.0 TOP: 5-6 Example 6 KEY: simplifying a complex number complex number multiplying complex numbers 10. ANS: B PTS: 1 DIF: L2 REF: 5-7 Completing the Square OBJ: Solving Equations by Completing the Square NAT: NAEP A4a CAT5.LV21/22.50 CAT5.LV21/22.52 IT.LV17/18.AM IT.LV17/18.CP S9.TSK3.NS S9.TSK3.PRA S10.TSK3.NS S10.TSK3.PRA TV.LV21/22.12 TV.LV21/22.52 TV.LVALG.53 TV.LVALG.57 STA: CA A2 8.0 TOP: 5-7 Example 1 KEY: perfect square trinomial 11. ANS: B PTS: 1 DIF: L2 REF: 5-7 Completing the Square OBJ: Solving Equations by Completing the Square NAT: NAEP A4a CAT5.LV21/22.50 CAT5.LV21/22.52 IT.LV17/18.AM IT.LV17/18.CP S9.TSK3.NS S9.TSK3.PRA S10.TSK3.NS S10.TSK3.PRA TV.LV21/22.12 TV.LV21/22.52 TV.LVALG.53 TV.LVALG.57 STA: CA A2 8.0 TOP: 5-7 Example 2 KEY: completing the square 12. ANS: D PTS: 1 DIF: L2 REF: 5-7 Completing the Square OBJ: Solving Equations by Completing the Square NAT: NAEP A4a CAT5.LV21/22.50 CAT5.LV21/22.52 IT.LV17/18.AM IT.LV17/18.CP S9.TSK3.NS S9.TSK3.PRA S10.TSK3.NS S10.TSK3.PRA TV.LV21/22.12 TV.LV21/22.52 TV.LVALG.53 TV.LVALG.57 STA: CA A2 8.0 TOP: 5-7 Example 3 KEY: completing the square 13. ANS: D PTS: 1 DIF: L2 REF: 5-7 Completing the Square OBJ: Solving Equations by Completing the Square NAT: NAEP A4a CAT5.LV21/22.50 CAT5.LV21/22.52 IT.LV17/18.AM IT.LV17/18.CP S9.TSK3.NS S9.TSK3.PRA S10.TSK3.NS S10.TSK3.PRA TV.LV21/22.12 TV.LV21/22.52 TV.LVALG.53 TV.LVALG.57 STA: CA A2 8.0 TOP: 5-7 Example 4 KEY: completing the square 14. ANS: D PTS: 1 DIF: L2 REF: 5-7 Completing the Square OBJ: Rewriting a Function by Completing the Square NAT: NAEP A4a CAT5.LV21/22.50 CAT5.LV21/22.52 IT.LV17/18.AM IT.LV17/18.CP S9.TSK3.NS S9.TSK3.PRA S10.TSK3.NS S10.TSK3.PRA TV.LV21/22.12 TV.LV21/22.52 TV.LVALG.53 TV.LVALG.57 STA: CA A2 8.0 TOP: 5-7 Example 6 KEY: vertex form quadratic equation 15. ANS: C PTS: 1 DIF: L2 REF: 5-8 The Quadratic Formula OBJ: Using the Quadratic Formula NAT: NAEP A4a NAEP A4c CAT5.LV21/22.50 CAT5.LV21/22.51 CAT5.LV21/22.53 IT.LV17/18.AM IT.LV17/18.CP IT.LV17/18.DI S9.TSK3.DSP S9.TSK3.NS S9.TSK3.PRA S10.TSK3.DSP S10.TSK3.NS S10.TSK3.PRA TV.LV21/22.12 TV.LV21/22.52 TV.LVALG.53 TV.LVALG.57 STA: CA A2 8.0 TOP: 5-8 Example 2 KEY: Quadratic Formula 2

12 16. ANS: B The zeros of the function are the x-intercepts of its graph. These are the solutions of the related quadratic equation because f(x) = 0 at those points. A B C D Feedback What are the x-intercepts of the graph? Correct! Find the zeros of the function, not the vertex. The zeros of the function are the solutions of the related equation. PTS: 1 DIF: Advanced REF: Page 298 OBJ: Solve quadratic equations by graphing. NAT: NA 1 NA 6 NA 9 NA 10 NA 2 STA: CA 8.0 TOP: Solve quadratic equations by graphing. KEY: Quadratic Equations Solve Quadratic Equations MSC: 1998 Lesson ANS: A To complete the square for any quadratic expression of the form x 2 + bx, find half of b, and square the result. Then, add the result to x 2 + bx. A B C D Feedback Correct! Did you make the quadratic expression a perfect square? Did you verify the answer by substituting the values? Did you check the signs of the roots? PTS: 1 DIF: Average REF: Page 311 OBJ: Solve quadratic equations by completing the square. NAT: NA 1 NA 3 NA 7 NA 10 NA 2 TOP: Solve quadratic equations by completing the square. KEY: Quadratic Equations Solve Quadratic Equations Completing the Square MSC: 1998 Lesson ANS: D To complete the square for any quadratic expression of the form x 2 + bx, find half of b, and square the result. Then, add the result to x 2 + bx. A B C D Feedback Did you make the quadratic expression a perfect square? Did you check the signs of the roots? Find both the solutions. Correct! PTS: 1 DIF: Average REF: Page 311 OBJ: Solve quadratic equations by completing the square. NAT: NA 1 NA 3 NA 7 NA 10 NA 2 TOP: Solve quadratic equations by completing the square. KEY: Quadratic Equations Solve Quadratic Equations Completing the Square MSC: 1998 Lesson 6-3 3

13 19. ANS: D The solution of a quadratic equation of the form ax 2 + bx + c = 0, where a 0, is obtained by using the formula x = b ± b 2 4ac 2a. A B C D Feedback Did you check the signs of the solution? Did you use the correct formula? Did you substitute the values of a, b, and c correctly in the formula? Correct! PTS: 1 DIF: Average REF: Page 318 OBJ: Solve quadratic equations by using the Quadratic Formula. NAT: NA 1 NA 6 NA 8 NA 9 NA 2 STA: CA 9.0 TOP: Solve quadratic equations by using the Quadratic Formula. KEY: Quadratic Equations Solve Quadratic Equations Quadratic Formula MSC: 1998 Lesson ANS: C If b 2 4ac > 0 and b 2 4ac is a perfect square, then the roots are rational. If b 2 4ac > 0 and b 2 4ac is not a perfect square, then the roots are real and irrational. A B C D Feedback Did you use the correct formula for the discriminant? Did you check the sign of the answer? Correct! Did you use the correct order of operations while evaluating the discriminant? PTS: 1 DIF: Basic REF: Page 318 OBJ: Use the discriminate to determine the number and type of roots of a quadratic equation. NAT: NA 1 NA 6 NA 8 NA 9 NA 2 STA: CA 9.0 TOP: Use the discriminate to determine the number and type of roots of a quadratic equation. KEY: Quadratic Equations Roots of Quadratic Equations Discriminates MSC: 1998 Lesson 6-4 4

14 21. ANS: C If b 2 4ac < 0, then the roots are complex. A B C D Feedback Did you use the correct order of operations while evaluating the discriminant? Did you use the correct formula for the discriminant? Correct! Did you check the sign of the answer? PTS: 1 DIF: Basic REF: Page 318 OBJ: Use the discriminate to determine the number and type of roots of a quadratic equation. NAT: NA 1 NA 6 NA 8 NA 9 NA 2 STA: CA 9.0 TOP: Use the discriminate to determine the number and type of roots of a quadratic equation. KEY: Quadratic Equations Roots of Quadratic Equations Discriminates MSC: 1998 Lesson ANS: A The vertex form of a quadratic function is y = a(x h) 2 + k. The equation of the axis of symmetry of a parabola is x = h. A B C D Feedback Correct! Did you check the x-coordinate of the vertex? Did you identify the coordinates of the vertex correctly? Did you use the correct equation of the axis of symmetry of a parabola? PTS: 1 DIF: Basic REF: Page 326 OBJ: Find the equation of the axis of symmetry of quadratic functions of the form y = a(x h)^2 + k. NAT: NA 2 NA 7 NA 8 NA 10 NA 6 STA: CA 9.0 TOP: Find the equation of the axis of symmetry of quadratic functions of the form y = a(x h)^2 + k. KEY: Quadratic Functions Axis of Symmetry MSC: 1998 Lesson ANS: A Complete the square of the function y = ax 2 + bx + c to write the function in the vertex form, y = a(x h) 2 + k. A B C D Feedback Correct! Did you verify the answer by graphing the function to find the location of its vertex? First, factor the coefficient from the quadratic and linear terms, and then complete the square and write in the vertex form. Did you write the quadratic function in the vertex form? PTS: 1 DIF: Average REF: Page 326 OBJ: Find the coordinates of the vertex of quadratic functions of the form y = a(x h)^2 + k. NAT: NA 2 NA 7 NA 8 NA 10 NA 6 TOP: Find the coordinates of the vertex of quadratic functions of the form y = a(x h)^2 + k. KEY: Quadratic Functions Vertex MSC: 1998 Lesson 6-6 5

15 SHORT ANSWER 24. ANS: PTS: 1 DIF: L2 REF: 5-3 Translating Parabolas OBJ: Using Vertex Form NAT: NAEP G2c CAT5.LV21/22.54 CAT5.LV21/22.56 IT.LV17/18.AM S9.TSK3.GM S9.TSK3.PRA S10.TSK3.GM S10.TSK3.PRA TV.LV21/22.14 TV.LV21/22.16 TV.LVALG.56 TV.LVALG.57 STA: CA A2 9.0 TOP: 5-3 Example 1 KEY: graphing translation parabola 25. ANS: two real solutions PTS: 1 DIF: L2 REF: 5-8 The Quadratic Formula OBJ: Using the Discriminant NAT: NAEP A4a NAEP A4c CAT5.LV21/22.50 CAT5.LV21/22.51 CAT5.LV21/22.53 IT.LV17/18.AM IT.LV17/18.CP IT.LV17/18.DI S9.TSK3.DSP S9.TSK3.NS S9.TSK3.PRA S10.TSK3.DSP S10.TSK3.NS S10.TSK3.PRA TV.LV21/22.12 TV.LV21/22.52 TV.LVALG.53 TV.LVALG.57 STA: CA A2 8.0 TOP: 5-8 Example 4 KEY: discriminant Quadratic Formula 6

16 ESSAY 26. ANS: a. The y-intercept is 5. The equation of axis of symmetry is x = 2. The x-coordinate of the vertex is 2. b. x x 2 + 4x 5 fx () Ê Ë Áx, fx () ˆ 4 ( 4) 2 + 4( 4) 5 5 ( 4, 5) 3 ( 3) 2 + 4( 3) 5 8 ( 3, 8) 2 ( 2) 2 + 4( 2) 5 9 ( 2, 9) 1 ( 1) 2 + 4( 1) 5 8 ( 1, 8) 0 (0) 2 + 4(0) 5 5 (0, 5) c. Assessment Rubric Level 3 Superior *Shows thorough understanding of concepts. *Uses appropriate strategies. *Computation is correct. *Written explanation is exemplary. *Diagram/table/chart is accurate (as applicable). *Goes beyond requirements of problem. 7

17 Level 2 Satisfactory *Shows understanding of concepts. *Uses appropriate strategies. *Computation is mostly correct. *Written explanation is effective. *Diagram/table/chart is mostly accurate (as applicable). *Satisfies all requirements of problem. Level 1 Nearly Satisfactory *Shows understanding of most concepts. *May not use appropriate strategies. *Computation is mostly correct. *Written explanation is satisfactory. *Diagram/table/chart is mostly accurate (as applicable). *Satisfies most of the requirements of problem. Level 0 Unsatisfactory *Shows little or no understanding of the concept. *May not use appropriate strategies. *Computation is incorrect. *Written explanation is not satisfactory. *Diagram/table/chart is not accurate (as applicable). *Does not satisfy requirements of problem. PTS: 1 DIF: Advanced REF: Page 343 OBJ: Solve problems and show solutions. NAT: NA 6 TOP: Solve problems and show solutions. KEY: Problem Solving Show Solutions 8

18 27. ANS: a. The graphs have the same vertex and the same shape. However, the graph of y = 1 3 x 2 opens up and the graph of y = 1 3 x 2 opens down. Changing the value of a in the equation y = a(x h) 2 + k can affect the direction of the opening and the shape of the graph. b. Both graphs open up and have the same shape. The vertex of each graph is on the x-axis. However, the graphs have different horizontal positions. Changing the value of h in the equation y = a(x h) 2 + k translates the graph horizontally. If h > 0, the graph translates h units to the right. If h < 0, the graph translates h units to the left. c. Both graphs have the same shape, and they open up. The vertex of each graph is on the y-axis. However, the graphs have different vertical positions. Changing the value of k in the equation y = a(x h) 2 + k translates the parabola along the y-axis. If k > 0, the parabola is translated k units up. If k < 0, it is translated k units down. Assessment Rubric Level 3 Superior *Shows thorough understanding of concepts. *Uses appropriate strategies. *Computation is correct. *Written explanation is exemplary. *Diagram/table/chart is accurate (as applicable). *Goes beyond requirements of problem. Level 2 Satisfactory *Shows understanding of concepts. *Uses appropriate strategies. *Computation is mostly correct. *Written explanation is effective. *Diagram/table/chart is mostly accurate (as applicable). *Satisfies all requirements of problem. Level 1 Nearly Satisfactory *Shows understanding of most concepts. *May not use appropriate strategies. *Computation is mostly correct. *Written explanation is satisfactory. *Diagram/table/chart is mostly accurate (as applicable). *Satisfies most of the requirements of problem. Level 0 Unsatisfactory *Shows little or no understanding of the concept. *May not use appropriate strategies. *Computation is incorrect. *Written explanation is not satisfactory. *Diagram/table/chart is not accurate (as applicable). 9

19 *Does not satisfy requirements of problem. PTS: 1 DIF: Advanced REF: Page 343 OBJ: Solve problems and show solutions. NAT: NA 6 TOP: Solve problems and show solutions. KEY: Problem Solving Show Solutions 28. ANS: [4] 2(x 3) = 2(x 2 6x + 9) + 5 = (2x 2 12x + 18) + 5 = 2x 2 12x + 23 When x = 3, the value of 2(x 3) is 2(0) or 5. If x is any number other than 3, then x 3 0 and (x 3) 2 is a positive number. So 2(x 3) 2 will also be a positive number. The sum of a positive number and 5 has to be greater than 5. Therefore, the value of 2(x 3) is 5 when x is 3 and greater than 5 when x is any number other than 3. Since 2(x 3) is equal to 2x 2 12x + 23, the minimum value of the function y = 2x 2 12x + 23 is 5. [3] Most of the reasoning is correct, but one or two points of the argument were not addressed thoroughly. [2] The reasoning was based too much on specific numerical values of x and y. [1] There were some correct observations, but there was no overall grasp of the situation. PTS: 1 DIF: L4 REF: 5-2 Properties of Parabolas OBJ: Finding Maximum and Minimum Values NAT: NAEP A2a CAT5.LV21/22.54 CAT5.LV21/22.56 IT.LV17/18.AM IT.LV17/18.PS S9.TSK3.GM S9.TSK3.PRA S10.TSK3.GM S10.TSK3.PRA TV.LV21/22.12 TV.LV21/22.14 TV.LV21/22.16 TV.LV21/22.17 TV.LVALG.56 TV.LVALG.57 STA: CA A2 9.0 CA A TOP: 5-2 Example 4 KEY: quadratic function minimum value writing in math reasoning extended response rubric-based question 10