Name: lass: ate: I: hapter 5 Test - lgebra 2 Honors /I-MYP Essay emonstrate your knowledge by giving a clear, concise solution to each problem. e sure to include all relevant drawings and justify your answers. You may show your solution in more than one way or investigate beyond the requirements of the problem. 1. Mr. Moseley asked the students in his lgebra class to work in groups to solve (x 3)2 = 25, stating that each student in the first group to solve the equation correctly would earn five bonus points on the next quiz. Mi-Ling s group solved the equation using the Square Root Property. Emilia s group used the Quadratic Formula to find the solutions. In which group would you prefer to be? Explain your reasoning. 4pts 2. The next day, Mr. Moseley had his students work in pairs to review for their chapter exam. He asked each student to write a practice problem for his or her partner. Len wrote the following problem for his partner, Jocelyn: Write an equation for the parabola whose vertex is ( 3, 4), that passes through ( 1, 0), and opens down. a. Jocelyn had trouble solving Len s problem. Explain why. b. How would you change Len s problem? c. Make the change you suggested in part b and complete the problem. 4pts 1
Name: I: 3. a. Write a quadratic function in vertex form whose maximum value is 8. b. Write a quadratic function that transforms the graph of your function from part a so that it is shifted horizontally. Explain the change you made and describe the transformation that results from this change. 4pts Short nswer Graph the quadratic inequality. 4. y < 2x 2-6x + 10 Points (ll lables includes vertex and axis and roots with coordinates 2 pt, correct parabolla graph 3pt, correct line type solid/broken 1pt, correct shade 2pt) 2
Name: I: Multiple hoice Identify the choice that best completes the statement or answers the question. 5. onsider the quadratic function f( x) = -2x 2 + 2x + 2. Find the y-intercept and the equation of the axis of symmetry. a. The y-intercept is 2. The equation of the axis of symmetry is x = - 1 2. b. The y-intercept is 1 2. The equation of the axis of symmetry is x = 2. c. The y-intercept is + 2. The equation of the axis of symmetry is x = 1 2. d. The y-intercept is - 1 2. The equation of the axis of symmetry is x = 2. 3
Name: I: 6. Graph the quadratic function f(x) = -2x 2 + 2x + 2. a. c. b. d. etermine whether the given function has a maximum or a minimum value. Then, find the maximum or minimum value of the function. 7. f(x) = x 2-2x + 2 a. The function has a maximum value. The maximum value of the function is 1. b. The function has a maximum value. The maximum value of the function is 5. c. The function has a minimum value. The minimum value of the function is 1. d. The function has a minimum value. The minimum value of the function is 5. 4
Name: I: 8. f(x) = -x 2 + 2x + 7 a. The function has a minimum value. The minimum value of the function is 8. b. The function has a minimum value. The minimum value of the function is 4. c. The function has a maximum value. The maximum value of the function is 4. d. The function has a maximum value. The maximum value of the function is 8. Solve the equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. 9. -x 2 + 4x = 0 a. c. b. The solution set is Ì 0, 4 Ó. d. The solution set is Ì -4, 0 Ó. The solution set is {-4 0}. The solution set is Ì 2, 4 Ó. 5
Name: I: 10. x 2 + 4x + 2 = 0 a. c. b. One solution is between 3 and 4, while the other solution is between 0 and 1. d. One solution is between 3 and 0, while the other solution is between 4 and 1. One solution is between 3 and 1, while the other solution is between 0 and 4. One solution is between 3 and 4, while the other solution is between 0 and 1. 6
Name: I: Write a quadratic equation with the given roots. Write the equation in the form ax 2 + bx + c = 0, where a, b, and c are integers. 11. - 5 4 and 8 a. 4x 2-27x - 40 = 0 c. x 2-27x - 40 = 0 b. 4x 2 + 27x + 40 = 0 d. x 2-27x + 40 = 0 Solve the equation by factoring. 12. 2x 2 + 3x - 14 = 0 a. { 4, - 7 } c. { 4, 7} 2 b. {- 7, 2} d. {2, 7} 2 Simplify. 13. 245 64 a. 7 5 8 c. 5 8 b. 49 8 d. 7 7 8 14. (2i)(-3i)(4i) a. -24 c. 24i b. -24i d. 24 15. i 7 a. -i c. i b. 1 d. -1 16. (-4 + 4i)(-3-3i) a. 16 + 12i c. 24 + 0i b. 12 + 0i - 12i 2 d. 12 + 0i + 12 17. 3 6 + 7i 18 a. 85 + 21 85 i c. 18 13 + 21 13 i 6 b. 85-7 85 i d. 18 85-21 85 i 7
Name: I: Solve the equation by using the Square Root Property. 18. 16x 2-48x + 36 = 49 a. { 3 } 2 c. 13 {- 4, 1 4 } b. { 3 2, 7} d. {- 1 4, 13 4 } 19. 100x 2-80x + 16 = 9 a. { 1 10, 7 10 } c. { 2 5 } b. {- 7 10, - 1 10 } d. { 2 5, 3} Solve the equation by completing the square. 20. x 2 + 2x - 3 = 0 a. Ì -3, 1 Ó b. Ì -6, 2 Ó c. Ì -6, 1 Ó d. Ì -1, 3 Ó Find the exact solution of the following quadratic equation by using the Quadratic Formula. 21. x 2-8x = 20 a. Ì -10, 2 Ó b. Ì 20, 28 Ó c. Ì -4, 20 Ó d. Ì -2, 10 Ó 22. -x 2 + 3x + 7 = 0 3-37 3 + 37 a. Ì, -2-2 Ó -3-12 -3 + 12 b. Ì, -2-2 Ó c. d. -3 - -19-3 + -19 Ì, -2-2 Ó -3-37 -3 + 37 Ì, -2-2 Ó 8
Name: I: Find the value of the discriminant. Then describe the number and type of roots for the equation. 23. -x 2-14x + 2 = 0 a. The discriminant is 196. ecause the discriminant is greater than 0 and is a perfect square, the two roots are real and rational. b. The discriminant is 204. ecause the discriminant is less than 0, the two roots are complex. c. The discriminant is 204. ecause the discriminant is greater than 0 and is not a perfect square, the two roots are real and irrational. d. The discriminant is 188. ecause the discriminant is less than 0, the two roots are complex. Write the following quadratic function in vertex form. Then, identify the axis of symmetry. 24. y = -3x 2 + 48x a. The vertex form of the function is y = 3 ( x + 8) 2 + 192. The equation of the axis of symmetry is x = -192. b. The vertex form of the function is y = ( x + 192) 2 + 8. The equation of the axis of symmetry is x = -8. c. The vertex form of the function is y = -3 ( x - 8) 2 + 192. The equation of the axis of symmetry is x = 8. d. The vertex form of the function is y = -3 ( x + 8) 2 + 192. The equation of the axis of symmetry is x = 192. 25. Write an equation for the parabola whose vertex is at Ê Ë Á 2, 6 ˆ and which passes through Ê ËÁ 4, - 1 ˆ. a. y = ( x + 2) 2-6 c. y = -1.75 ( x - 2) 2 + 6 b. y = 1.75 ( x - 2) 2 + 6 d. y = -1.75 ( x + 2) 2-6 9
I: hapter 5 Test - lgebra 2 Honors /I-MYP nswer Section ESSY 1. NS: Student responses should indicate that using the Square Root Property, as Mi-Ling s group did, would take less time than the other method since the equation is already set up as a perfect square set equal to a constant. To solve using the other method, the binomial would need to be expanded and the constant on the right brought to the left side of the equal sign. PTS: 4 2. NS: a. Jocelyn had trouble because the problem is impossible. No such parabola exists. b. Student responses will vary. One of the three conditions must be omitted or modified. Sample answer:...that passes through ( 1, 12). c. nswers will vary and depend on the answer for part b. For example, for the sample answer in part b above, a possible equation is: y = 2(x + 3) 2 4. PTS: 4 3. NS: a. nswer must be of the form y = a(x h) 2 + 8 where h is any real number and a < 0. b. nswers must be of the form y = a[x (h + n)] 2 + 8 where h and a represent the same values as in part a. The student choice is for the value of n. The student should indicate that the graph will shift to the left n units if his or her value of n is negative, but will shift the graph to the right n units if the chosen value of n is positive. PTS: 4 1
I: SHORT NSWER 4. NS: Graph the related quadratic equation. Since the inequality symbol is <, the parabola should be dashed. Test a point (x 1, y 1 ) inside the parabola. If (x 1, y 1 ) is the solution of the inequality, shade the region inside the parabola. If (x 1, y 1 ) is not a solution, shade the region outside the parabola. PTS: 4 IF: dvanced REF: Lesson 5-8 OJ: 5-8.1 Graph quadratic inequalities in two variables. TOP: Graph quadratic inequalities in two variables. KEY: Quadratic Inequalities Graph Quadratic Inequalities ST: M.912..4.1.1 M.912..10.3 MULTIPLE HOIE 5. NS: For the quadratic equation ax 2 + bx + c, the y-intercept is c and the equation of axis of symmetry is x = -b 2a. id you check the signs? id you interchange the y-intercept and the x-coordinate of the vertex? id you use the correct formulas for the y-intercept and the x-coordinate of the vertex? PTS: 4 IF: verage REF: Lesson 5-1 OJ: 5-1.1 Graph quadratic functions. ST: M.912..2.6 M.912..7.6 M.912..10.3 TOP: Graph quadratic functions. KEY: Quadratic Functions Graph Quadratic Functions 2
I: 6. NS: First, choose integer values for x. Then evaluate the function for each x value. Graph the resulting coordinate pairs and connect the points with a smooth curve. Graph ordered pairs that satisfy the function. id you plot the graph correctly? When the coefficient of x 2 is less than 0, the graphs opens down. PTS: 4 IF: dvanced REF: Lesson 5-1 OJ: 5-1.1 Graph quadratic functions. ST: M.912..2.6 M.912..7.6 M.912..10.3 TOP: Graph quadratic functions. KEY: Quadratic Functions Graph Quadratic Functions 7. NS: The y-coordinate of the vertex of a quadratic function is the maximum or minimum value obtained by the function. The coefficient of x 2 is greater than zero. The graph of this function opens up. What is the value of the y-coordinate of the vertex? PTS: 4 IF: verage REF: Lesson 5-1 OJ: 5-1.2 Find and interpret the maximum and minimum values of a quadratic function. ST: M.912..2.6 M.912..7.6 M.912..10.3 TOP: Find and interpret the maximum and minimum values of a quadratic function. KEY: Maximum Values Minimum Values Quadratic Functions 8. NS: The y-coordinate of the vertex of a quadratic function is the maximum or minimum value obtained by the function. The graph of the function opens down. The coefficient of x 2 is less than zero. What is the value of the y-coordinate of the vertex? PTS: 4 IF: verage REF: Lesson 5-1 OJ: 5-1.2 Find and interpret the maximum and minimum values of a quadratic function. ST: M.912..2.6 M.912..7.6 M.912..10.3 TOP: Find and interpret the maximum and minimum values of a quadratic function. KEY: Maximum Values Minimum Values Quadratic Functions 3
I: 9. NS: 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. The zeros of the function are the solutions of the related equation. What are the x-intercepts of the graph? Find the zeros of the function, not the vertex. PTS: 4 IF: dvanced REF: Lesson 5-2 OJ: 5-2.1 Solve quadratic equations by graphing. ST: M.912..7.6 M.912..7.10 TOP: Solve quadratic equations by graphing. KEY: Quadratic Equations Solve Quadratic Equations 10. NS: When exact roots cannot be found by graphing, you can estimate solutions by stating the consecutive integers between which the roots are located. Is the coefficient of x 2 less than zero? id you graph the function correctly? When the coefficient of x 2 is greater than 0, the graph opens up. PTS: 4 IF: dvanced REF: Lesson 5-2 OJ: 5-2.2 Estimate solutions of quadratic equations by graphing. ST: M.912..7.6 M.912..7.10 TOP: Estimate solutions of quadratic equations by graphing. KEY: Quadratic Equations Solve Quadratic Equations 11. NS: quadratic equation with roots p and q can be written as (x - p)(x - q) = 0, which can be further simplified. id you check the signs of the coefficients? id you calculate the coefficients correctly? id you verify the answer by substituting the values? PTS: 4 IF: verage REF: Lesson 5-3 OJ: 5-3.1 Write quadratic equations in intercept form. TOP: Write quadratic equations in intercept form. KEY: Quadratic Equations Roots of Quadratic Equations ST: M.912..4.3 M.912..10.3 4
I: 12. NS: For any real numbers a and b, if ab = 0, then either a = 0, b = 0, or both a and b are equal to zero. id you use the Zero Product Property correctly? id you factor the binomial correctly? id you verify the answer by substituting the values? PTS: 4 IF: verage REF: Lesson 5-3 OJ: 5-3.2 Solve quadratic equations by factoring. ST: M.912..4.3 M.912..10.3 TOP: Solve quadratic equations by factoring. KEY: Quadratic Equations Solve Quadratic Equations Factoring 13. NS: a b = a b heck the numerator. heck the square root of the numerator. heck your calculation. PTS: 4 IF: verage REF: Lesson 5-4 OJ: 5-4.1 Find square roots. ST: M.912..1.6 TOP: Find square roots. 14. NS: Multiply the real numbers and imaginary numbers separately. heck your calculation. heck the sign. Multiply the imaginary numbers again. PTS: 4 IF: verage REF: Lesson 5-4 OJ: 5-4.2 Perform operations with pure imaginary numbers. ST: M.912..1.6 TOP: Perform operations with pure imaginary numbers. 5
I: 15. NS: Multiply the real numbers and imaginary numbers separately. heck your calculation. heck the sign. ompute again. PTS: 4 IF: verage REF: Lesson 5-4 OJ: 5-4.2 Perform operations with pure imaginary numbers. ST: M.912..1.6 TOP: Perform operations with pure imaginary numbers. 16. NS: Use the FOIL method to multiply the complex numbers and use the formula i 2 = -1. ombine the real parts and then the imaginary parts of the two numbers. Use the FOIL method to find the product. Use the value of i 2. ombine the real parts. PTS: 4 IF: verage REF: Lesson 5-4 OJ: 5-4.4 Perform multiplication operations with complex numbers. ST: M.912..1.6 TOP: Perform multiplication operations with complex numbers. KEY: omplex Numbers Multiply omplex Numbers 17. NS: Multiply the numerator as well as the denominator by the conjugate of the denominator. Use the FOIL method and the difference of squares to simplify the given expression. Multiply the numerator with the conjugate of the denominator. Have you multiplied the constant in the numerator with its conjugate of the denominator? id you multiply the conjugates correctly in the denominator? PTS: 4 IF: verage REF: Lesson 5-4 OJ: 5-4.5 Perform division operations with complex numbers. ST: M.912..1.6 TOP: Perform division operations with complex numbers. KEY: omplex Numbers ivide omplex Numbers 6
I: 18. NS: For any real number n, if x 2 = n, then x = ± n. id you use the Square Root Property correctly? id you verify the answer by substituting the values? id you factor the perfect square trinomial correctly? PTS: 4 IF: verage REF: Lesson 5-5 OJ: 5-5.1 Solve quadratic equations by using the Square Root Property. ST: M.912..7.3 M.912..7.5 TOP: Solve quadratic equations by using the Square Root Property. KEY: Quadratic Equations Solve Quadratic Equations Square Root Property 19. NS: For any real number n, if x 2 = n, then x = ± n. id you factor the perfect square trinomial correctly? id you use the Square Root Property correctly? id you verify the answer by substituting the values? PTS: 4 IF: verage REF: Lesson 5-5 OJ: 5-5.1 Solve quadratic equations by using the Square Root Property. ST: M.912..7.3 M.912..7.5 TOP: Solve quadratic equations by using the Square Root Property. KEY: Quadratic Equations Solve Quadratic Equations Square Root Property 20. NS: 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. id you make the quadratic expression a perfect square? id you verify the answer by substituting the values? id you check the signs of the roots? PTS: 4 IF: verage REF: Lesson 5-5 OJ: 5-5.2 Solve quadratic equations by completing the square. ST: M.912..7.3 M.912..7.5 TOP: Solve quadratic equations by completing the square. KEY: Quadratic Equations Solve Quadratic Equations ompleting the Square 7
I: 21. NS: 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. id you check the signs of the solution? id you use the correct formula? id you substitute the values of a, b, and c correctly in the formula? PTS: 4 IF: verage REF: Lesson 5-6 OJ: 5-6.1 Solve quadratic equations by using the Quadratic Formula. ST: M.912..7.4 M.912..7.5 M.912..10.3 TOP: Solve quadratic equations by using the Quadratic Formula. KEY: Quadratic Equations Solve Quadratic Equations Quadratic Formula 22. NS: 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. id you substitute the values of a, b, and c correctly in the formula? id you evaluate the discriminant correctly? id you use the correct formula? PTS: 4 IF: verage REF: Lesson 5-6 OJ: 5-6.1 Solve quadratic equations by using the Quadratic Formula. ST: M.912..7.4 M.912..7.5 M.912..10.3 TOP: Solve quadratic equations by using the Quadratic Formula. KEY: Quadratic Equations Solve Quadratic Equations Quadratic Formula 8
I: 23. NS: 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. id you use the correct formula for the discriminant? id you check the sign of the answer? id you use the correct order of operations while evaluating the discriminant? PTS: 4 IF: asic REF: Lesson 5-6 OJ: 5-6.2 Use the discriminant to determine the number and types of roots of a quadratic equation. ST: M.912..7.4 M.912..7.5 M.912..10.3 TOP: Use the discriminant to determine the number and types of roots of a quadratic equation. KEY: Quadratic Equations Roots of Quadratic Equations iscriminates 24. NS: 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. id you use the correct equation of the axis of symmetry? id you check the x-coordinate of the vertex? id you identify the coordinates of the vertex correctly? PTS: 4 IF: asic REF: Lesson 5-7 OJ: 5-7.1 nalyze quadratic functions in the form y = a(x - h)^2 + k. ST: M.912..2.10 TOP: nalyze quadratic functions in the form y = a(x - h)^2 + k. KEY: Quadratic Functions xis of Symmetry 25. NS: If the vertex and another point on the graph of a parabola are known, the equation of the parabola can be written in vertex form. id you substitute correctly in the vertex form of the equation? id you find the correct coefficient values? id you check the signs of the coefficients? PTS: 4 IF: verage REF: Lesson 5-7 OJ: 5-7.2 Write a quadratic function in the form y = a(x - h)^2 + k. ST: M.912..2.10 TOP: Write a quadratic function in the form y = a(x h)^2 + k. KEY: Quadratic Functions 9