Algebra 2 10/23/14 Review for Test 2 (non-calculator section) What you’ll learn and why… I can learn how to solve the problems in the Practice Test so I can ace Test 2 on Monday. HW: Know the formulas and methods needed to solve each problem in the Practice Test. Warm-up: Silently recite the Quadratic Formula 5x. Silently recite the formula to find the x-coordinate of the vertex of a quadratic equation in standard form 5x.

1. What is the product of 𝟒+𝟐𝒊 and 𝟒−𝟐𝒊 written in standard form? 20 16 + 4i D. 16 – 4i Notes: 4+2𝑖 and 4−2𝑖 are complex conjugates. The product of complex conjugates is always a real number. From this information, choices C and D should be eliminated. Rule or Formula to multiply complex conjugates: 𝒂+𝒃𝒊 𝒂−𝒃𝒊 = 𝒂 𝟐 + 𝒃 𝟐 𝟒+𝟐𝒊 𝟒−𝟐𝒊 = 𝟒 𝟐 + 𝟐 𝟐 =𝟏𝟔+𝟒=𝟐𝟎 The answer is B 20. Your Turn: Answer Practice Test #1.

Your Turn: Answer Practice Test #2 2. Simplify 𝟑+𝟒𝒊 −(−𝟐−𝟓𝒊). A. −𝟏+𝟗𝒊 B. 𝟏−𝒊 C. 𝟓+𝟗𝒊 D. 𝟓−𝒊 Notes: To subtract a complex number, add its opposite. One Method: 3+4𝑖 − −2−5𝑖 = 3+4𝑖 +(2+5𝑖) =3+4𝑖+2+5𝑖 =𝟓+𝟗𝒊 Answer: C Your Turn: Answer Practice Test #2

3. Find the domain and range of each function. B D: R: C D: R: D D: R: All real numbers All real numbers All real numbers All real numbers 𝒚≥𝟎 𝒚 ≤ 𝟎 𝒚 ≤ 1 𝒚≥−𝟐 Notes: The domain of a function is the set of all possible x values. The range of a function is the set of all possible y values.

Your Turn: Answer Practice Test #4. 4. Which of the following is not true of the function 𝒇 𝒙 = 𝒙 𝟐 −𝟐𝒙+𝟓? 2. Since 𝑎>0, the parabola opens up, hence the function has a minimum. The minimum is the y-coordinate of the vertex. To find the minimum, first find the x-coordinate of the vertex. Use the formula for the axis of symmetry to find the x-coordinate, 𝒙= −𝒃 𝟐𝒂 The y-intercept is 5. The minimum value is 4. The axis of symmetry is 𝒙=𝟏. The vertex is at (1, 5). Notes: 1. The function 𝑓 is in standard form, 𝑓 𝑥 =𝑎 𝑥 2 +𝑏𝑥+𝑐. 2. The y-intercept is at x=0, so y−intercept=𝑐=5. Hence, Choice A is true. Since a = 1 and b = -2, 𝒙= −(−𝟐) 𝟐(𝟏) =𝟏 3. To find the y-coordinate of the vertex, substitute the value of x into the original function. 𝒚= (𝟏) 𝟐 −𝟐 𝟏 +𝟓=𝟏−𝟐+𝟓=−𝟏+𝟓=𝟒 Hence, Choices B and C are true. The statement which is not true is D. The vertex is at (1, 4) Your Turn: Answer Practice Test #4.

Your Turn: Answer Practice Test #5 5. If 𝒙 𝟐 −𝟔𝒙+𝟐= 𝒙−𝒉 𝟐 +𝒌, what is the value of 𝒌? -7 -11 7 11 Notes: 1. 𝑥−ℎ 2 +𝑘 is the vertex form of a quadratic expression. 2. In the vertex form, (h, k) is the vertex and 𝑘 is the y-coordinate . Possible method: Find the coordinates of the vertex by using the formula for the axis of symmetry or the x-coordinate: 𝒙= −𝒃 𝟐𝒂 = −(−𝟔) 𝟐(𝟏) = 𝟔 𝟐 =𝟑 Next, solve for the y-coordinate of the vertex by substituting 𝑥=3 into the original expression. 𝒚= (𝟑) 𝟐 −𝟔 𝟑 +𝟐=𝟗−𝟏𝟖+𝟐=−𝟗+𝟐=−𝟕 Since 𝒚=−𝟕, 𝒕𝒉𝒆𝒓𝒆𝒇𝒐𝒓𝒆 𝒌=−𝟕. The correct answer is A. Alternative Method: You could use completing the square to convert from standard form to vertex form. Your Turn: Answer Practice Test #5

6. Find the roots of 𝟏𝟔 𝒙 𝟐 +𝟑=𝟎. A. 𝑥=± 3 4 Solution: Goal: Rewrite the equation so that it is in the form 𝒙 𝟐 =𝒄. 𝒙=± 𝒄 B. 𝑥=± −3 16 C. 𝑥=± 𝑖 3 4 16 𝑥 2 +3=0 D. 𝑥=± 3 4 16 𝑥 2 =−3 Subtract 3 from each side 𝑥 2 = −3 16 Divide each side by 16. Note: Eliminate answers that are not in simplest form. Choices A and B are eliminated because they have a radical in the denominator. The choices left are C and D. C has nonreal solutions while D has real solutions. 𝑥=± −3 16 𝒙 𝟐 =𝒄. 𝒙=± 𝒄 𝑥=± −3 16 𝐐𝐮𝐨𝐭𝐢𝐞𝐧𝐭 𝐑𝐮𝐥𝐞 𝐨𝐟 𝐑𝐚𝐝𝐢𝐜𝐚𝐥𝐬 𝒎 𝒏 = 𝒎 𝒏 𝒙=± 𝒊 𝟑 𝟒 Simplify.

7. Factor 𝟐𝟓 𝒙 𝟐 +𝟔𝟒. Notes: 25 𝑥 2 +64 is a Sum of Two Squares. The factors of a Sum of Two Squares are two complex conjugates. Eliminate Choices A and B. A. 5𝑥+8 (5𝑥−8) B. (5𝑥+8)(5𝑥+8) C. (5𝑥−8𝑖)(5𝑥+8𝑖) D. (5𝑥+8𝑖)(5𝑥+8𝑖) The rule is: 𝒂 𝟐 + 𝒃 𝟐 =(𝒂+𝒃𝒊)(𝒂−𝒃𝒊) 𝟐𝟓𝒙 𝟐 +𝟔𝟒=(𝟓𝒙+𝟖𝒊)(𝟓𝒙−𝟖𝒊) The answer is C. Alternative Method: Work Backwards Start from the answer choices. Use FOIL to find which factors will multiply to get the product 𝟐𝟓𝒙 𝟐 +𝟔𝟒. 𝟓𝒙−𝟖𝒊 𝟓𝒙+𝟖𝒊 =𝟐𝟓 𝒙 𝟐 +𝟒𝟎𝒙𝒊−𝟒𝟎𝒙𝒊−𝟔𝟒 𝒊 𝟐 =𝟐𝟓 𝒙 𝟐 −𝟔𝟒 −𝟏 =𝟐𝟓 𝒙 𝟐 +𝟔𝟒 Your Turn: Answer Practice Test #7.

8. Solve 𝒙 𝟐 +𝟖𝟏=𝟎 over the set of complex numbers. A. ±9 B. ±9𝑖 C. ±81 D. ±81𝑖 Notes: The goal is to rewrite the equation in the form: 𝒙 𝟐 =𝒄 𝒙=± 𝒄 𝑥 2 +81=0 𝑥 2 =−81 Subtract 81 from each side. 𝑥=± −81 𝒙 𝟐 =𝒄 𝒙=± 𝒄 𝑥=±9𝑖 Simplify. 𝒊= −𝟏 The answer is B. Your Turn: Answer Practice Test #8.

Your Turn: Answer Practice Test #9. Function A and Function B are continuous quadratic functions. Function A Function B 𝒇 𝒙 = 𝒙 𝟐 +𝟒𝒙−𝟓 Which function has a greater negative x-intercept? Function A Function B The x-intercepts are equal. Solution: You need to find the negative x-intercepts of both function and then compare to find the greater number. The negative x-intercept of Function B is -1. Use factoring to find the x-intercepts of Function A. (Other methods may be used.) 𝒙 𝟐 +𝟒𝒙−𝟓=𝟎 𝒂=𝟏, 𝒃=𝟒, 𝒄=−𝟓 𝒙+𝟓 𝒙−𝟏 =𝟎 Look for factors of c whose sum is b. Use 5 and -1 𝒙+𝟓=𝟎 𝒐𝒓 𝒙−𝟏=𝟎 Set each factor equal to 0. 𝒙=−𝟓 𝒐𝒓 𝒙=𝟏 Solve for x. The negative x-intercept of Function A is -5. Since −𝟏>−𝟓, the answer is B. Your Turn: Answer Practice Test #9.

10. Solve the equation 𝟓 𝒙 𝟐 +𝟓𝒙=𝟔𝟎 by factoring. 𝒙=−𝟒 𝒐𝒓 𝒙=𝟑 𝒙=𝟒 𝒐𝒓 𝒙=−𝟑 Steps: 𝟓 𝒙 𝟐 +𝟓𝒙=𝟔𝟎 𝒙 𝟐 +𝒙=𝟏𝟐 Divide each side by the GCF, 5. 𝒙 𝟐 +𝒙−𝟏𝟐=𝟎 Subtract 12 from each side. (𝒙+𝟒)(𝒙−𝟑)=𝟎 Look for factors of -12 whose sum is 1. Use 4 and -3. 𝒙+𝟒=𝟎 𝒐𝒓 𝒙−𝟑=𝟎 Set each factor equal to 0. 𝒙=−𝟒 𝒐𝒓 𝒙=𝟑 Solve for x. The answer is C. Alternative Method: Work Backwards Substitute the answer choices into the given equation. See which solutions satisfy the given equation. Your Turn: Answer Practice Test #10.

Your Turn: Answer Practice Test #11. 11. Which of the following functions has its vertex below the x-axis? A. 𝒇 𝒙 =− 𝒙 𝟐 B. 𝒇 𝒙 =−𝟓 (𝒙+𝟒) 𝟐 +𝟑 C. 𝒇 𝒙 = 𝒙 𝟐 +𝟐 D. 𝒇 𝒙 =𝟐 (𝒙−𝟏) 𝟐 −𝟐 Solution: To determine which vertex is below the x-axis, graph the vertex of each function. Vertex Form: 𝑓 𝑥 =𝑎 (𝑥−ℎ) 2 +𝑘 Vertex is at (h, k) Vertex of A: (0, 0) Vertex of C: (0, 2) Vertex of B: (-4, 3) Vertex of D: (1, -2) The answer is D. Is there a pattern? What pattern do you see? Given the vertex (h, k), when k < 0, then the vertex is below the x-axis. Your Turn: Answer Practice Test #11.

12. What is the equation of the parabola shown? Notes: 1. The parabola opens up, therefore a > 0. B. 𝑓 𝑥 =− 1 4 𝑥 2 C. 𝑓 𝑥 = 1 4 𝑥 2 Eliminate Choices A and B. D. 𝑓 𝑥 = 4𝑥 2 2. USE NICE POINTS! The point (1, 4) is on the parabola. Check by substituting x = 1 and y = 4 into the remaining equations C and D. C. 𝑓 1 = 1 4 (1) 2 = 1 4 D. 𝑓 1 =4 (1) 2 = 4 The answer is D. Your Turn: Answer Practice Test #12.

13. What is the solution set of 𝟑 𝒙 𝟐 +𝟖𝒙>−𝟑𝒙−𝟔? B. −3≤𝑥≤− 2 3 C. 𝑥<−3 𝑜𝑟 𝑥>− 2 3 D. 𝑥>−3 𝑜𝑟 𝑥>− 2 3

13. What is the solution set of 𝟑 𝒙 𝟐 +𝟖𝒙>−𝟑𝒙−𝟔? B. −3≤𝑥≤− 2 3 C. 𝑥<−3 𝑜𝑟 𝑥>− 2 3 D. 𝑥>−3 𝑜𝑟 𝑥>− 2 3 STEPS 1. Write the original inequality. 𝟑 𝒙 𝟐 +𝟖𝒙>−𝟑𝒙−𝟔 2. Set one side of the inequality to 0. (Add 3x and 6 to each side.) 𝟑 𝒙 𝟐 +𝟖𝒙+𝟑𝒙+𝟔>𝟎 3. Combine like terms. 𝟑 𝒙 𝟐 +𝟏𝟏𝒙+𝟔>𝟎 4. Change the inequality sign to equal sign. 𝟑 𝒙 𝟐 +𝟏𝟏𝒙+𝟔=𝟎 5. Solve the quadratic equation to find the zeros or x-intercepts. 𝒙=−𝟑 𝒐𝒓 𝒙=− 𝟐 𝟑 6. Mark the x-intercepts on a numberline. The numberline is now divided into three intervals: A, B, and C. Sketch the parabola. 7. Determine in which interval/s the solutions lie. Write the solution set using the correct inequality symbols. 8. Check by using test values. 𝐴𝑡 𝑥=−4: 3 −4 2 +11 −4 +6=48−44+6=10 10>0 𝐴𝑡 𝑥=0: 3 (0) 2 +11 0 +6=6 6>0 A B C − 𝟐 𝟑 −𝟑 The solutions lie in intervals A and C. 𝒙<−𝟑 𝒐𝒓 𝒙 >− 𝟐 𝟑

14. Which of the following quadratic equations has no real roots? B. 𝟓 𝒙 𝟐 −𝟑𝒙−𝟒=𝟎 Write in standard form. 𝑎=5, 𝑏=−3, 𝑐=−4 5 𝑥 2 +3𝑥−4=0 𝑏 2 −4𝑎𝑐= −3 2 −4 5 −4 >𝟎 𝑎=5, 𝑏=3, 𝑐=−4 + + 𝑏 2 −4𝑎𝑐= 3 2 −4 5 −4 >𝟎 + + D. 𝟓 𝒙 𝟐 +𝟑𝒙=−𝟒 C. 𝟓 𝒙 𝟐 =𝟑𝒙 Write in standard form. Write in standard form. 5 𝑥 2 +3𝑥+4=0 5 𝑥 2 −3𝑥=0 𝑎=5, 𝑏=−3, 𝑐=0 𝑎=5, 𝑏=3, 𝑐=4 𝑏 2 −4𝑎𝑐= 3 2 −4 5 4 <𝟎 𝑏 2 −4𝑎𝑐= −3 2 −4 5 0 >𝟎 - + + + Note: Use the discriminant, 𝒃 𝟐 −𝟒𝒂𝒄, to find out which quadratic equation has no real roots. If 𝒃 𝟐 −𝟒𝒂𝒄 > 0, the equation has 2 real solutions. If 𝒃 𝟐 −𝟒𝒂𝒄 = 0, the equation has 1 real solution. If 𝒃 𝟐 −𝟒𝒂𝒄 < 0, the equation has 2 nonreal solutions. 9 80

Your Turn: Answer Practice Test #15. 15. Find the vertex of 𝒚= 𝟐𝒙 𝟐 −𝟒𝒙+𝟓 and state if it is a maximum or minimum. Notes: Since a > 0, the parabola opens up and it has a minimum value. (1, 3); maximum (1, 3); minimum (3, 1); maximum (3, 1); minimum Eliminate Choices A and C. To find the x-coordinate of the vertex, use the formula 𝒙= −𝒃 𝟐𝒂 . 𝒙= −𝒃 𝟐𝒂 = −(−𝟒) 𝟐(𝟐) = 𝟒 𝟒 =𝟏 The answer is B. Your Turn: Answer Practice Test #15.