Download presentation
Presentation is loading. Please wait.
Published byJoy Lang Modified over 9 years ago
1
Algebra 2 10/23/14 Review for Test 2 (non-calculator section)
What youโll learn and whyโฆ 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.
2
1. What is the product of ๐+๐๐ and ๐โ๐๐ written in standard form?
20 16 + 4i D. 16 โ 4i Notes: 4+2๐ and 4โ2๐ 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: ๐+๐๐ ๐โ๐๐ = ๐ ๐ + ๐ ๐ ๐+๐๐ ๐โ๐๐ = ๐ ๐ + ๐ ๐ =๐๐+๐=๐๐ The answer is B 20. Your Turn: Answer Practice Test #1.
3
Your Turn: Answer Practice Test #2
2. Simplify ๐+๐๐ โ(โ๐โ๐๐). A. โ๐+๐๐ B. ๐โ๐ C. ๐+๐๐ D. ๐โ๐ Notes: To subtract a complex number, add its opposite. One Method: 3+4๐ โ โ2โ5๐ = 3+4๐ +(2+5๐) =3+4๐+2+5๐ =๐+๐๐ Answer: C Your Turn: Answer Practice Test #2
4
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 ๐โฅ๐ ๐ โค ๐ ๐ โค 1 ๐โฅโ๐ 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.
5
Your Turn: Answer Practice Test #4.
4. Which of the following is not true of the function ๐ ๐ = ๐ ๐ โ๐๐+๐? 2. Since ๐>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, ๐= โ๐ ๐๐ The y-intercept is 5. The minimum value is 4. The axis of symmetry is ๐=๐. The vertex is at (1, 5). Notes: 1. The function ๐ is in standard form, ๐ ๐ฅ =๐ ๐ฅ 2 +๐๐ฅ+๐. 2. The y-intercept is at x=0, so yโintercept=๐=5. Hence, Choice A is true. Since a = 1 and b = -2, ๐= โ(โ๐) ๐(๐) =๐ 3. To find the y-coordinate of the vertex, substitute the value of x into the original function. ๐= (๐) ๐ โ๐ ๐ +๐=๐โ๐+๐=โ๐+๐=๐ 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.
6
Your Turn: Answer Practice Test #5
5. If ๐ ๐ โ๐๐+๐= ๐โ๐ ๐ +๐, what is the value of ๐? -7 -11 7 11 Notes: 1. ๐ฅโโ 2 +๐ is the vertex form of a quadratic expression. 2. In the vertex form, (h, k) is the vertex and ๐ 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: ๐= โ๐ ๐๐ = โ(โ๐) ๐(๐) = ๐ ๐ =๐ Next, solve for the y-coordinate of the vertex by substituting ๐ฅ=3 into the original expression. ๐= (๐) ๐ โ๐ ๐ +๐=๐โ๐๐+๐=โ๐+๐=โ๐ Since ๐=โ๐, ๐๐๐๐๐๐๐๐๐ ๐=โ๐. 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
7
6. Find the roots of ๐๐ ๐ ๐ +๐=๐. A. ๐ฅ=ยฑ 3 4 Solution:
Goal: Rewrite the equation so that it is in the form ๐ ๐ =๐. ๐=ยฑ ๐ B. ๐ฅ=ยฑ โ3 16 C. ๐ฅ=ยฑ ๐ 3 4 16 ๐ฅ 2 +3=0 D. ๐ฅ=ยฑ 16 ๐ฅ 2 =โ3 Subtract 3 from each side ๐ฅ 2 = โ3 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. ๐ฅ=ยฑ โ3 16 ๐ ๐ =๐. ๐=ยฑ ๐ ๐ฅ=ยฑ โ ๐๐ฎ๐จ๐ญ๐ข๐๐ง๐ญ ๐๐ฎ๐ฅ๐ ๐จ๐ ๐๐๐๐ข๐๐๐ฅ๐ฌ ๐ ๐ = ๐ ๐ ๐=ยฑ ๐ ๐ ๐ Simplify.
8
7. Factor ๐๐ ๐ ๐ +๐๐. Notes: 25 ๐ฅ is a Sum of Two Squares. The factors of a Sum of Two Squares are two complex conjugates. Eliminate Choices A and B. A. 5๐ฅ+8 (5๐ฅโ8) B. (5๐ฅ+8)(5๐ฅ+8) C. (5๐ฅโ8๐)(5๐ฅ+8๐) D. (5๐ฅ+8๐)(5๐ฅ+8๐) The rule is: ๐ ๐ + ๐ ๐ =(๐+๐๐)(๐โ๐๐) ๐๐๐ ๐ +๐๐=(๐๐+๐๐)(๐๐โ๐๐) 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 ๐๐๐ ๐ +๐๐. ๐๐โ๐๐ ๐๐+๐๐ =๐๐ ๐ ๐ +๐๐๐๐โ๐๐๐๐โ๐๐ ๐ ๐ =๐๐ ๐ ๐ โ๐๐ โ๐ =๐๐ ๐ ๐ +๐๐ Your Turn: Answer Practice Test #7.
9
8. Solve ๐ ๐ +๐๐=๐ over the set of complex numbers.
A. ยฑ9 B. ยฑ9๐ C. ยฑ81 D. ยฑ81๐ Notes: The goal is to rewrite the equation in the form: ๐ ๐ =๐ ๐=ยฑ ๐ ๐ฅ 2 +81=0 ๐ฅ 2 =โ81 Subtract 81 from each side. ๐ฅ=ยฑ โ81 ๐ ๐ =๐ ๐=ยฑ ๐ ๐ฅ=ยฑ9๐ Simplify. ๐= โ๐ The answer is B. Your Turn: Answer Practice Test #8.
10
Your Turn: Answer Practice Test #9.
Function A and Function B are continuous quadratic functions. Function A Function B ๐ ๐ = ๐ ๐ +๐๐โ๐ 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.) ๐ ๐ +๐๐โ๐=๐ ๐=๐, ๐=๐, ๐=โ๐ ๐+๐ ๐โ๐ =๐ Look for factors of c whose sum is b. Use 5 and -1 ๐+๐=๐ ๐๐ ๐โ๐=๐ Set each factor equal to 0. ๐=โ๐ ๐๐ ๐=๐ Solve for x. The negative x-intercept of Function A is -5. Since โ๐>โ๐, the answer is B. Your Turn: Answer Practice Test #9.
11
10. Solve the equation ๐ ๐ ๐ +๐๐=๐๐ by factoring.
๐=โ๐ ๐๐ ๐=๐ ๐=๐ ๐๐ ๐=โ๐ Steps: ๐ ๐ ๐ +๐๐=๐๐ ๐ ๐ +๐=๐๐ Divide each side by the GCF, 5. ๐ ๐ +๐โ๐๐=๐ Subtract 12 from each side. (๐+๐)(๐โ๐)=๐ Look for factors of -12 whose sum is 1. Use 4 and -3. ๐+๐=๐ ๐๐ ๐โ๐=๐ Set each factor equal to 0. ๐=โ๐ ๐๐ ๐=๐ 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.
12
Your Turn: Answer Practice Test #11.
11. Which of the following functions has its vertex below the x-axis? A. ๐ ๐ =โ ๐ ๐ B. ๐ ๐ =โ๐ (๐+๐) ๐ +๐ C. ๐ ๐ = ๐ ๐ +๐ D. ๐ ๐ =๐ (๐โ๐) ๐ โ๐ Solution: To determine which vertex is below the x-axis, graph the vertex of each function. Vertex Form: ๐ ๐ฅ =๐ (๐ฅโโ) 2 +๐ 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.
13
12. What is the equation of the parabola shown?
Notes: 1. The parabola opens up, therefore a > 0. B. ๐ ๐ฅ =โ 1 4 ๐ฅ 2 C. ๐ ๐ฅ = 1 4 ๐ฅ 2 Eliminate Choices A and B. D. ๐ ๐ฅ = 4๐ฅ 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. ๐ 1 = 1 4 (1) 2 = 1 4 D. ๐ 1 =4 (1) 2 = 4 The answer is D. Your Turn: Answer Practice Test #12.
14
13. What is the solution set of ๐ ๐ ๐ +๐๐>โ๐๐โ๐?
B. โ3โค๐ฅโคโ 2 3 C. ๐ฅ<โ3 ๐๐ ๐ฅ>โ 2 3 D. ๐ฅ>โ3 ๐๐ ๐ฅ>โ 2 3
15
13. What is the solution set of ๐ ๐ ๐ +๐๐>โ๐๐โ๐?
B. โ3โค๐ฅโคโ 2 3 C. ๐ฅ<โ3 ๐๐ ๐ฅ>โ 2 3 D. ๐ฅ>โ3 ๐๐ ๐ฅ>โ 2 3 STEPS 1. Write the original inequality. ๐ ๐ ๐ +๐๐>โ๐๐โ๐ 2. Set one side of the inequality to 0. (Add 3x and 6 to each side.) ๐ ๐ ๐ +๐๐+๐๐+๐>๐ 3. Combine like terms. ๐ ๐ ๐ +๐๐๐+๐>๐ 4. Change the inequality sign to equal sign. ๐ ๐ ๐ +๐๐๐+๐=๐ 5. Solve the quadratic equation to find the zeros or x-intercepts. ๐=โ๐ ๐๐ ๐=โ ๐ ๐ 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. ๐ด๐ก ๐ฅ=โ4: 3 โ โ4 +6=48โ44+6= >0 ๐ด๐ก ๐ฅ=0: 3 (0) = >0 A B C โ ๐ ๐ โ๐ The solutions lie in intervals A and C. ๐<โ๐ ๐๐ ๐ >โ ๐ ๐
16
14. Which of the following quadratic equations has no real roots?
B. ๐ ๐ ๐ โ๐๐โ๐=๐ Write in standard form. ๐=5, ๐=โ3, ๐=โ4 5 ๐ฅ 2 +3๐ฅโ4=0 ๐ 2 โ4๐๐= โ3 2 โ4 5 โ4 >๐ ๐=5, ๐=3, ๐=โ4 + + ๐ 2 โ4๐๐= โ4 5 โ4 >๐ + + D. ๐ ๐ ๐ +๐๐=โ๐ C. ๐ ๐ ๐ =๐๐ Write in standard form. Write in standard form. 5 ๐ฅ 2 +3๐ฅ+4=0 5 ๐ฅ 2 โ3๐ฅ=0 ๐=5, ๐=โ3, ๐=0 ๐=5, ๐=3, ๐=4 ๐ 2 โ4๐๐= โ4 5 4 <๐ ๐ 2 โ4๐๐= โ3 2 โ4 5 0 >๐ - + + + Note: Use the discriminant, ๐ ๐ โ๐๐๐, to find out which quadratic equation has no real roots. If ๐ ๐ โ๐๐๐ > 0, the equation has 2 real solutions. If ๐ ๐ โ๐๐๐ = 0, the equation has 1 real solution. If ๐ ๐ โ๐๐๐ < 0, the equation has 2 nonreal solutions. 9 80
17
Your Turn: Answer Practice Test #15.
15. Find the vertex of ๐= ๐๐ ๐ โ๐๐+๐ 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 ๐= โ๐ ๐๐ . ๐= โ๐ ๐๐ = โ(โ๐) ๐(๐) = ๐ ๐ =๐ The answer is B. Your Turn: Answer Practice Test #15.
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.