Algebra 2 Midterm Review

Transcription

1 Name: Class: Date: Algebra 2 Midterm Review Short Answer 1. Find the product (2x 3y) Find the zeros of f(x) = x 2 + 7x + 9 by using the Quadratic Formula. 3. Solve the polynomial equation 2x 5 + 8x 4 6x 3 = 0 by factoring. 4. Rewrite the polynomial 9x x 5 + 8x 2 x 4 9x in standard form. Then, identify the leading coefficient, degree, and number of terms. Name the polynomial. 5. Factor x 3 + 5x 2 4x Find the values of x and y that make the equation 10x + 10i = 30 (30y)i true. 7. Find the product ( 5x 3)(x 3 5x + 2). 8. Find the roots of the equation 42x 63 = 7x 2 by factoring. 9. Using the graph of f(x) = x 2 as a guide, describe the transformations, and then graph the function g(x) = (x 3) Add. Write the result in the form a + bi. (1 5i) + (9 + 7i) 11. Identify the degree of the monomial 4r 4 s. 12. Find the absolute value 4 + 6i. 13. Find the number and type of solutions for x 2 + 6x = Find the minimum or maximum value of f(x) = x 2 + 3x 4. Then state the domain and range of the function. 15. Let g( x) be a vertical shift of f( x) = x up 4 units followed by a vertical stretch by a factor of 3. Write the rule for g( x). 16. What expression is equivalent to (3 2i) 2? 17. Find the product 3a 2 b 2 (a 4 b 3 + 5a b 2 ). 1

2 Name: 18. Identify the parent function for g( x) = ( x + 3) 3 and describe what transformation of the parent function it represents. 19. Identify the axis of symmetry for the graph of f(x) = x 2 + 2x Graph the complex number 3 + 2i. 21. Factor the expression 81x x 5 y Solve the equation x 2 = 35 2x by completing the square. 23. Express 3 78 in terms of i. 24. Graph the function f(x) = x 3 + 3x 2 6x Solve the equation 2x = Determine whether the binomial (x 2) is a factor of the polynomial P( x) = x 3 x 2 4x Use synthetic substitution to evaluate the polynomial P( x) = x 3 2x 2 + 2x 5 for x = Find the zeros of g( x) = 4x 2 x + 4 by using the Quadratic Formula. 29. The parent function f(x) = x 2 is reflected across the x-axis, vertically stretched by a factor of 10, and translated right 10 units to create g. Use the description to write the quadratic function in vertex form. 30. Find the zeros of the function h( x) = x 2 24x + 80 by factoring. 31. Divide by using synthetic division. (x 2 4x + 8) ( x 2) 32. Complete the square for the expression x x +. Write the resulting expression as a binomial squared. 33. Consider the function f(x) = 4x 2 8x Determine whether the graph opens upward or downward. Find the axis of symmetry, the vertex and the y-intercept. Graph the function. 34. Find the zeros of f( x) = x 2 + 4x 5 by using a graph and table. 35. Write the function f(x) = 6x 2 24x 25 in vertex form, and identify its vertex. 2

3 Name: 36. Add. Write your answer in standard form. (4e 4 e 2 ) + (e 4 + 5e 2 4) 37. Identify the roots of 5x 3 35x x = 0. State the multiplicity of each root. 38. Simplify 3 + 4i 3 3i. 39. Use Pascal s Triangle to expand the expression ( 3x + 2) Identify all of the real roots of 4x x 3 4x 2 89x + 22 = Find the complex conjugate of 12i Using the graph of f(x) = x 2 as a guide, describe the transformations, and then graph the function g(x) = 7x Write a quadratic function in standard form with zeros 8 and Graph f(x) = x 2 5x + 6 by using a table. 45. Simplify 15i Solve the equation x 2 12x + 36 = Multiply 6i( 4 3i). Write the result in the form a + bi. 48. Divide by using long division: (5x + 6x 3 8) (x 2). 49. Find the zeros of the function f(x) = x 2 4x

4 Algebra 2 Midterm Review Answer Section SHORT ANSWER 1. ANS: 8x 3 36x 2 y + 54xy 2 27y 3 Write in expanded form. (2x 3y)(2x 3y)(2x 3y) Multiply the last two binomial factors. (2x 3y)(4x 2 12xy + 9y 2 ) Distribute the first term, distribute the second term, and combine like terms. 8x 3 36x 2 y + 54xy 2 27y 3 PTS: 1 DIF: Average REF: 15e53b6e df-9c7d f0d2ea OBJ: Expanding a Power of a Binomial NAT: NT.CCSS.MTH A.APR.1 TOP: 3-2 Multiplying Polynomials 2. ANS: 7 ± 13 x = 2 x 2 + 7x + 9 = 0 Set f(x) = 0. x = b ± b 2 4ac 2a x = 7 ± (7)2 4(1)(9) 2(1) x = x = 7 ± ± 13 2 Write the Quadratic Formula. Substitute 1 for a, 7 for b, and 9 for c. Simplify. Write in simplest form. PTS: 1 DIF: Average REF: 15837ab df-9c7d f0d2ea OBJ: Quadratic Functions with Real Zeros NAT: NT.CCSS.MTH A.REI.4 STA: DC.DCLS.MTH.05.AL2.AII.P.5 DC.DCLS.MTH.05.AL2.AII.P.6 DC.DCLS.MTH.05.AL2.AII.P.7 LOC: MTH.C MTH.C TOP: 2-6 The Quadratic Formula KEY: quadratic formula 1

5 3. ANS: The roots are 0, 3, and 1. 2x 5 + 8x 4 6x 3 = 0 Factor out the GCF, 2x 3. 2x 3 Ê x 2 ˆ 4x + 3 Ë Á = 0 2x 3 ( x 3) ( x 1) = 0 Factor the quadratic. 2x 3 = 0, x 3 = 0, x 1 = 0 Set each factor equal to 0. x = 0, x = 3, x = 1 Solve for x. PTS: 1 DIF: Average REF: 15f8273e df-9c7d f0d2ea OBJ: Using Factoring to Solve Polynomial Equations NAT: NT.CCSS.MTH A.APR.2 NT.CCSS.MTH A.APR.3 STA: DC.DCLS.MTH.05.AL2.AII.P.7 LOC: MTH.C TOP: 3-5 Finding Real Roots of Polynomial Equations 4. ANS: 9x 5 x 4 9x 3 + 8x 2 9x + 11 leading coefficient: 9; degree: 5; number of terms: 6; name: quintic polynomial The standard form is written with the terms in order from highest to lowest degree. In standard form, the degree of the first term is the degree of the polynomial. The polynomial has 6 terms. It is a quintic polynomial. PTS: 1 DIF: Average REF: 15d df-9c7d f0d2ea OBJ: Classifying Polynomials LOC: MTH.C MTH.C MTH.C TOP: 3-1 Polynomials MSC: DOK 2 5. ANS: (x + 5)(x 2)(x + 2) (x 3 + 5x 2 ) + ( 4x 20) Group terms. = x 2 (x + 5) 4(x + 5) Factor common monomials from each group. = (x + 5)(x 2 4) Factor out the common binomial. = (x + 5)(x 2)(x + 2) Factor the difference of squares. PTS: 1 DIF: Average REF: 15f1273a df-9c7d f0d2ea OBJ: Factoring by Grouping NAT: NT.CCSS.MTH A.SSE.2 LOC: MTH.C TOP: 3-4 Factoring Polynomials 2

6 6. ANS: x = 3, y = x = 30 Equate the real parts. x = 3 Solve for x. 10 = 30y Equate the imaginary parts. 1 3 = y Solve for y. PTS: 1 DIF: Average REF: 157eb5fa df-9c7d f0d2ea OBJ: Equating Two Complex Numbers LOC: MTH.C TOP: 2-5 Complex Numbers and Roots KEY: complex numbers 7. ANS: 5x 4 3x 3 25x x 6 ( 5x 3)(x 3 5x + 2) = 5x(x 3 5x + 2) 3(x 3 5x + 2) Distribute 5x and 3. = 5x(x 3 ) + 5x( 5x) + 5x( 2) 3(x 3 ) 3( 5x) 3( 2) Distribute 5x and 3 again. = 5x 4 25x x 3x x 6 Multiply. = 5x 4 3x 3 25x x 6 Combine like terms. PTS: 1 DIF: Average REF: 15e2b df-9c7d f0d2ea OBJ: Multiplying Polynomials NAT: NT.CCSS.MTH A.APR.1 LOC: MTH.C TOP: 3-2 Multiplying Polynomials 8. ANS: x = 3 Rewrite the equation in standard form, factor out the GCF, and then factor the perfect square trinomial. 42x 63 = 7x 2 7x 2 42x + 63 = 0 7(x 2 6x + 9) = 0 (x 2 6x + 9) = 0 (x 3) 2 = 0 x 3 = 0 x = 3 PTS: 1 DIF: Average REF: 1572a31e df-9c7d f0d2ea OBJ: Finding Roots by Using Special Factors NAT: NT.CCSS.MTH A.REI.4 TOP: 2-3 Solving Quadratic Equations by Graphing and Factoring KEY: solve quadratic equations 3

7 9. ANS: g(x) is f(x) translated 3 units right and 5 units down. Because h = 3, the graph is translated 3 units right. Because k = 5, the graph is translated 5 units down. Therefore, g(x) is f(x) translated 3 units right and 5 units down. PTS: 1 DIF: Average REF: 155fb74e df-9c7d f0d2ea OBJ: Translating Quadratic Functions NAT: NT.CCSS.MTH F.BF.3 STA: DC.DCLS.MTH.05.AL2.AII.P.11 LOC: MTH.C MTH.C MTH.C TOP: 2-1 Using Transformations to Graph Quadratic Functions KEY: quadratic graph 4

8 10. ANS: i To add complex numbers, add the real parts and the imaginary parts. To subtract complex numbers, subtract the real parts and the imaginary parts. (1 5i) + (9 + 7i) = (1 + (9)) + (7 + ( 6))i = i PTS: 1 DIF: Average REF: 159db4a df-9c7d f0d2ea OBJ: Adding and Subtracting Complex Numbers NAT: NT.CCSS.MTH N.CN.2 STA: DC.DCLS.MTH.05.AL2.AII.N.3 LOC: MTH.C TOP: 2-9 Operations with Complex Numbers 11. ANS: 5 Add the exponents of the variables = 5 The degree is 5. PTS: 1 DIF: Basic REF: 15d6ed df-9c7d f0d2ea OBJ: Identifying the Degree of a Monomial LOC: MTH.C TOP: 3-1 Polynomials MSC: DOK ANS: 2 13 (4) 2 2 Find the square root of the sum of the squares of the real and + (+ 6) imaginary parts of the complex number Simplify the square root. PTS: 1 DIF: Basic REF: 159d8d df-9c7d f0d2ea OBJ: Determining the Absolute Value of Complex Numbers NAT: NT.CCSS.MTH N.CN.3 LOC: MTH.C TOP: 2-9 Operations with Complex Numbers 13. ANS: The equation has one real solution. x 2 + 6x = 9 Make sure the equation is in standard form, ax 2 + bx + c = 0. x 2 + 6x + 9 = 0 b 2 4ac Evaluate the discriminant. = ( 6) 2 4( 1) ( 9) =36 36 = 0 The discriminant is zero. The equation has one real solution. PTS: 1 DIF: Average REF: a df-9c7d f0d2ea OBJ: Analyzing Quadratic Equations by Using the Discriminant NAT: NT.CCSS.MTH A.REI.4 LOC: MTH.C MTH.C TOP: 2-6 The Quadratic Formula KEY: quadratic formula 5

9 14. ANS: The minimum value is D: {all real numbers}; R: {y y 6.25} Step 1 Determine whether the function has a minimum or maximum value. Because a is positive, the graph opens upward and has a minimum value. Step 2 Find the x-value of the vertex. x = b = 3 = 1.5 2a 2(1) Step 3 Find the y-value of the vertex. f( 1.5) = ( 1.5) 2 + 3( 1.5) 4 = 6.25 The minimum value is The domain is {all real numbers}. The range is {y y 6.25}. PTS: 1 DIF: Average REF: be df-9c7d f0d2ea OBJ: Finding Minimum or Maximum Values NAT: NT.CCSS.MTH F.IF.7 STA: DC.DCLS.MTH.05.AL2.AII.P.5 LOC: MTH.C MTH.C TOP: 2-2 Properties of Quadratic Functions in Standard Form KEY: maximum minimum 15. ANS: g( x) = 3x + 12 Step 1 Translating f( x) = x up 4 units adds 4 to the function: Let h( x) = f( x) + 4 Substitute x for f( x) : h( x) = x + 4 Step 2 Stretching vertically by a factor of 3 multiplies the function by 3: g( x) = 3h( x) Substitute x + 4 for h( x): g( x) = 3( x + 4) Simplify: g( x) = 3x + 12 PTS: 1 DIF: Average REF: 14784b df-9c7d f0d2ea OBJ: Combining Transformations of Linear Functions NAT: NT.CCSS.MTH A.CED.2 NT.CCSS.MTH F.BF.3 STA: DC.DCLS.MTH.05.AL2.AII.P.11 LOC: MTH.C TOP: 1-3 Transforming Linear Functions KEY: transform linear functions shift translate stretch MSC: DOK 4 6

10 16. ANS: 5 12i (3 2i) 2 = (3 2i)(3 2i) = 9 6i 6i + 4i 2 Multiply. = 9 12i 4 Combine like terms. i 2 = 1. = 5 12i Simplify. PTS: 1 DIF: Advanced REF: 15a df-9c7d f0d2ea NAT: NT.CCSS.MTH N.CN.2 STA: DC.DCLS.MTH.05.AL2.AII.N.3 LOC: MTH.C TOP: 2-9 Operations with Complex Numbers 17. ANS: 3a 6 b a 3 b 4 Use the Distributive Property to multiply the monomial by each term inside the parentheses. Group terms to get like bases together, and then multiply. PTS: 1 DIF: Basic REF: 15e076b df-9c7d f0d2ea OBJ: Multiplying a Monomial and a Polynomial NAT: NT.CCSS.MTH A.APR.1 LOC: MTH.C TOP: 3-2 Multiplying Polynomials 18. ANS: The parent function is the cubic function, f( x) = x 3. g( x) = ( x + 3) 3 represents a horizontal translation of the parent function 3 units to the left. f( x) = x 3 intersects the x-axis at the point (0,0), and g( x) = ( x + 3) 3 intersects the x-axis at the point ( 3, 0). Thus, g( x) represents a horizontal translation of the cubic parent function 3 units to the left. PTS: 1 DIF: Average REF: 140f df-9c7d f0d2ea OBJ: Identifying Transformations of Parent Functions NAT: NT.CCSS.MTH F.BF.3 STA: DC.DCLS.MTH.05.AL2.AII.P.11 LOC: MTH.C MTH.C TOP: 1-2 Introduction to Parent Functions 7

11 19. ANS: x = 1 If a function has one zero, use the x-coordinate of the vertex to find the axis of symmetry. If a function has two zeros, use the average of the two zeros to find the axis of symmetry. PTS: 1 DIF: Average REF: 1566b df-9c7d f0d2ea OBJ: Identifying the Axis of Symmetry NAT: NT.CCSS.MTH F.IF.7 STA: DC.DCLS.MTH.05.AL2.AII.P.5 LOC: MTH.C TOP: 2-2 Properties of Quadratic Functions in Standard Form 20. ANS: The real axis is the x-axis, and the imaginary axis is the y-axis. Think of a + bi as x + yi. Thus the complex number 3 + 2i is at ( 3, 2). PTS: 1 DIF: Basic REF: 159b2b3a df-9c7d f0d2ea OBJ: Graphing Complex Numbers NAT: NT.CCSS.MTH N.CN.4 LOC: MTH.C TOP: 2-9 Operations with Complex Numbers KEY: graph complex number MSC: DOK 2 8

12 21. ANS: 3x 5 (3x + 4y)(9x 2 12xy + 16y 2 ) Factor out the GCF. 3x 5 (27x y 3 ) Write as a sum of cubes. 3x 5 ((3x) 3 + (4y) 3 ) Factor. 3x 5 (3x + 4y)((3x) 2 12xy + (4y) 2 ) = 3x 5 (3x + 4y)(9x 2 12xy + 16y 2 ) PTS: 1 DIF: Basic REF: 15f df-9c7d f0d2ea OBJ: Factoring the Sum or Difference of Two Cubes NAT: NT.CCSS.MTH A.SSE.2 LOC: MTH.C TOP: 3-4 Factoring Polynomials 22. ANS: x = 5 or x = 7 x 2 = 35 2x x 2 + 2x = 35 Collect variable terms on one side. x 2 Ê + 2x Ë Á ˆ 2 Ê = Ë Á ˆ 2 Ê b Add 2 Ë Á ˆ 2 to each side. x 2 + 2x + 1 = 36 Simplify. (x + 1) 2 = 36 Factor. x + 1 = ±6 Take the square root of each side. x + 1 = 6 or x + 1 = 6 Solve for x. x = 5 or x = 7 PTS: 1 DIF: Basic REF: 1579ca df-9c7d f0d2ea OBJ: Solving a Quadratic Equation by Completing the Square NAT: NT.CCSS.MTH A.REI.4 STA: DC.DCLS.MTH.05.AL2.AII.P.6 DC.DCLS.MTH.05.AL2.AII.P.7 LOC: MTH.C TOP: 2-4 Completing the Square KEY: complete the square solve quadratic equations 9

13 23. ANS: 3i = 3 ( 1)(78) Factor out 1. = Product Property = Simplify. = 3i 78 Express in terms of i. PTS: 1 DIF: Average REF: 157c2c8e df-9c7d f0d2ea OBJ: Simplifying Square Roots of Negative Numbers NAT: NT.CCSS.MTH N.CN.1 STA: DC.DCLS.MTH.05.AL2.AII.N.3 LOC: MTH.C MTH.C TOP: 2-5 Complex Numbers and Roots KEY: complex numbers MSC: DOK 2 10

14 24. ANS: Step 1: Identify the possible rational roots by using the Rational Root Theorem. p = 8 and q = 1, so roots are positive and negative values in multiples of 2 from 1 to 8. Step 2: Test possible rational zeros until a zero is identified. Test x = Test x = x = 1 is a zero, and f(x) = (x + 1)(x 2 + 2x 8). Step 3: Factor: f(x) = (x + 1)(x 2)(x + 4). The zeros are 1, 2, and 4. Step 4: Plot other points as guidelines. f(0) = 8 so the y-intercept is 8. Plot points between the zeros. f(1) = 10 and f( 3) = 10 Step 5: Identify end behavior. The degree is odd and the leading coefficient is positive, so as x, P(x) and as x +, P(x) +. Step 6: Sketch the graph by using all of the information about f(x). PTS: 1 DIF: Average REF: 1608fed df-9c7d f0d2ea OBJ: Graphing Polynomial Functions NAT: NT.CCSS.MTH A.APR.3 NT.CCSS.MTH F.IF.7.c STA: DC.DCLS.MTH.05.AL2.AII.P.5 LOC: MTH.C TOP: 3-7 Investigating Graphs of Polynomial Functions 11

15 25. ANS: x = ±6i 2x = 0 2x 2 = 72 Add 72 to both sides. x 2 = 36 Divide both sides by 2. x = ± 36 Take square roots. x = ±6i Express in terms of i. PTS: 1 DIF: Average REF: 157e8eea df-9c7d f0d2ea OBJ: Solving a Quadratic Equation with Imaginary Solutions NAT: NT.CCSS.MTH A.REI.4 STA: DC.DCLS.MTH.05.AL2.AII.P.6 DC.DCLS.MTH.05.AL2.AII.P.7 LOC: MTH.C MTH.C TOP: 2-5 Complex Numbers and Roots KEY: complex numbers 26. ANS: (x 2) is not a factor of the polynomial P( x) = x 3 x 2 4x + 3. Find P( 2) by synthetic substitution Since P( 2) 0, x 2 is not a factor of the polynomial P( x) = x 3 x 2 4x + 3. PTS: 1 DIF: Average REF: 15f1002a df-9c7d f0d2ea OBJ: Determining Whether a Linear Binomial is a Factor NAT: NT.CCSS.MTH A.SSE.2 LOC: MTH.C TOP: 3-4 Factoring Polynomials 27. ANS: P( 2) = 1 Write the coefficients of the dividend. Use a = P( 2) = 1 PTS: 1 DIF: Basic REF: 15ee9dce df-9c7d f0d2ea OBJ: Using Synthetic Substitution LOC: MTH.C TOP: 3-3 Dividing Polynomials 12

16 28. ANS: x = 1 ± 63 i = 4x 2 x + 4 Set g( x) = 0. x = b ± b 2 4ac 2a x = ( 1) ± ( 1) 2 4( 4) ( 4) 2( 4) x = 1 ± x = 1 ± i ± 63 i 8 = = 1 ± 63 8 Write the Quadratic Formula. Substitute 4 for a, 1 for b and 4 for c. Simplify. Write in terms of i. PTS: 1 DIF: Average REF: 1585b5fe df-9c7d f0d2ea OBJ: Quadratic Functions with Complex Zeros NAT: NT.CCSS.MTH A.REI.4 NT.CCSS.MTH N.CN.7 STA: DC.DCLS.MTH.05.AL2.AII.P.5 DC.DCLS.MTH.05.AL2.AII.P.6 DC.DCLS.MTH.05.AL2.AII.P.7 LOC: MTH.C MTH.C TOP: 2-6 The Quadratic Formula KEY: quadratic formula 29. ANS: g(x) = 10(x 10) 2 Identify how each transformation affects the coefficients in vertex form. reflection across x-axis a is negative. verticalstretch by a factor of 10 a = 10 translated right 10 units h = 10 Write the transformed function, using the vertex form g(x) = a(x h) 2 + k. g(x) = 10(x 10) Substitute 10 for a, 10 for h, and 0 for k. g(x) = 10(x 10) 2 Simplify. PTS: 1 DIF: Average REF: f df-9c7d f0d2ea OBJ: Writing Transformed Quadratic Functions NAT: NT.CCSS.MTH F.BF.3 STA: DC.DCLS.MTH.05.AL2.AII.P.5 LOC: MTH.C TOP: 2-1 Using Transformations to Graph Quadratic Functions KEY: quadratic graph parent function 13

17 30. ANS: x = 20 or x = 4 h( x) = x 2 24x + 80 x 2 24x + 80 = 0 Set the function equal to 0. (x 20)(x 4) = 0 Factor: Find factors of 80 that add to 24. x 20 = 0 or x 4 = 0 Apply the Zero-Product Property. x = 20 or x = 4 Solve each equation. PTS: 1 DIF: Basic REF: c df-9c7d f0d2ea OBJ: Finding Zeros by Factoring NAT: NT.CCSS.MTH A.REI.4 NT.CCSS.MTH F.IF.8 STA: DC.DCLS.MTH.05.AL2.AII.P.5 LOC: MTH.C TOP: 2-3 Solving Quadratic Equations by Graphing and Factoring KEY: solve quadratic equations 31. ANS: x x 2 For ( x 2), a = Write the coefficients of the expression Write the remainder as a fraction to get x x 2. Bring down the first coefficient. Multiply and add each column. PTS: 1 DIF: Average REF: 15ec3b df-9c7d f0d2ea OBJ: Using Synthetic Division to Divide by a Linear Binomial NAT: NT.CCSS.MTH A.APR.6 LOC: MTH.C TOP: 3-3 Dividing Polynomials 32. ANS: ( x + 5) 2 Ê 10ˆ 2 Ë Á = ( 5) Ê b = 25 Find 2 Ë Á ˆ 2. x x + 25 Add. ( x + 5) Factor. PTS: 1 DIF: Average REF: d df-9c7d f0d2ea OBJ: Completing the Square NAT: NT.CCSS.MTH A.REI.4 STA: DC.DCLS.MTH.05.AL2.AII.P.6 DC.DCLS.MTH.05.AL2.AII.P.7 LOC: MTH.C TOP: 2-4 Completing the Square KEY: complete the square solve quadratic equations 14

18 33. ANS: The parabola opens downward. The axis of symmetry is the line x = 1. The vertex is the point ( 1,14). The y-intercept is 10. Because a is 2, the graph opens downward. The axis of symmetry is given by x = ( 8) 2( 4) = 8 8 = 1. x = 1 is the axis of symmetry. The vertex lies on the axis of symmetry, so x = 1. The y-value is the value of the function at this x-value. f( 1) = 4( 1) 2 8( 1) + 10 = = 14 The vertex is ( 1,14). Because the last term is 10, the y-intercept is 10. PTS: 1 DIF: Average REF: ae df-9c7d f0d2ea OBJ: Graphing Quadratic Functions in Standard Form NAT: NT.CCSS.MTH F.IF.7 STA: DC.DCLS.MTH.05.AL2.AII.P.5 DC.DCLS.MTH.05.AL2.AII.G.3 LOC: MTH.C TOP: 2-2 Properties of Quadratic Functions in Standard Form 15

19 34. ANS: 5 and 1 Graph the function f( x) = x 2 + 4x 5. The graph opens upward because 1 > 0. The y-intercept is 5 because c = 5. Find the vertex: x = b = + 4 = 2 2a 2( 1) Find f( 2) : f( x) = x 2 + 4x 5 f( 2) = ( 2) 2 + 4( 2) 5 Substitute 2 for x. f( 2) = 9 The vertex is ( 2, 9). Plot the vertex and the y-intercept. Use symmetry and a table of values to find additional points. x f( x) A zero of a function is the x-value that makes the function equal to 0. The x-intercept is the same as the zero of a function because it s the value of x when y = 0. Look at the graph and find the x-intercepts. PTS: 1 DIF: Average REF: 156e df-9c7d f0d2ea OBJ: Finding Zeros by Using a Graph or Table NAT: NT.CCSS.MTH A.REI.11 STA: DC.DCLS.MTH.05.AL2.AII.P.5 LOC: MTH.C TOP: 2-3 Solving Quadratic Equations by Graphing and Factoring KEY: solve quadratic equations 16

20 35. ANS: f(x) = 6(x + 2) 2 1; vertex: ( 2, 1) f(x) = 6x 2 Factor to make the coefficient of the first 24x 25 term 1. f(x) = 6(x 2 + 4x +? ) 25? Set up to complete the square. f(x) = 6(x 2 Ê + 4x Ë Á ˆ 2 Ê 4 ) 25 ( 6) 2 Ë Á ˆ 2 Ê b Add and subtract 2 Ë Á ˆ 2. f(x) = 6(x + 2) 2 25 ( 6)( 2) 2 Simplify and factor. f(x) = 6(x + 2) 2 1 Simplify. The equation is now in vertex form, f(x) = a( x h) 2 + k, and the vertex is (h, k) or ( 2, 1). PTS: 1 DIF: Average REF: 1579f df-9c7d f0d2ea OBJ: Writing a Quadratic Function in Vertex Form NAT: NT.CCSS.MTH F.IF.4 STA: DC.DCLS.MTH.05.AL2.AII.P.5 LOC: MTH.C MTH.C TOP: 2-4 Completing the Square 36. ANS: 5e 4 + 4e 2 4 (4e 4 e 2 ) + (e 4 + 5e 2 4) = (4e 4 + 5e 2 ) + ( e 2 + e 4 ) + ( 4) Identify like terms. Rearrange terms to get like terms together. = 5e 4 + 4e 2 4 Combine like terms. PTS: 1 DIF: Basic REF: 15db8aee df-9c7d f0d2ea OBJ: Adding and Subtracting Polynomials NAT: NT.CCSS.MTH A.APR.1 LOC: MTH.C TOP: 3-1 Polynomials MSC: DOK ANS: x 5 is a factor once, and x + 6 is a factor twice. The root 5 has a multiplicity of 1, and the root 6 has a multiplicity of 2. 5x 3 35x x = 0 5x 3 35x x = 5( x 5) ( x + 6) ( x + 6) x 5 is a factor once, and x + 6 is a factor twice. The root 5 has a multiplicity of 1. The root 6 has a multiplicity of 2. PTS: 1 DIF: Average REF: 15fa899a df-9c7d f0d2ea OBJ: Identifying Multiplicity NAT: NT.CCSS.MTH A.APR.3 TOP: 3-5 Finding Real Roots of Polynomial Equations 17

21 38. ANS: i 3 + 4i 3 3i = ( 3 + 4i) ( 3 3i) (3 + 3i) (3 + 3i) Multiply by the conjugate. 9 9i + 12i + 12i2 = 9 + 9i 9i 9i 2 Distribute i 12 = Use i 2 = i 6 6 Simplify. PTS: 1 DIF: Average REF: 15a4b4aa df-9c7d f0d2ea OBJ: Dividing Complex Numbers NAT: NT.CCSS.MTH N.CN.3 STA: DC.DCLS.MTH.05.AL2.AII.N.3 LOC: MTH.C TOP: 2-9 Operations with Complex Numbers 39. ANS: 81x x x x + 16 The coefficients for n = 4 or row 5 of Pascal s Triangle are 1, 4, 6, 4, and 1. ( 3x + 2) È = 1( 3x) ( + 2) ÎÍ È 4( 3x) 3 ( + 2) ÎÍ È 6( 3x) 2 ( + 2) ÎÍ È 4( 3x) 1 ( + 2) ÎÍ È 1( 3x) 0 ( + 2) ÎÍ = 81x x x x + 16 PTS: 1 DIF: Average REF: 15e776ba df-9c7d f0d2ea OBJ: Using Pascal's Triangle to Expand Binomial Expressions NAT: NT.CCSS.MTH A.APR.5 TOP: 3-2 Multiplying Polynomials 18

22 40. ANS: 2, 1 4, , The possible rational roots are ±1, ± 1, ± Test The remainder is 0, so 2 is a root. Now test The remainder is 0, so 1 4 is a root. The polynomial factors to (x + 2)(x 1 4 )(4x x 44)., ± 2, ± 11, ± 11 2, ± To find the remaining roots, solve 4x x 44 = 0. Factor out the common factor to get 4(x 2 + 6x 11) = 0. Use the quadratic formula to find the irrational roots. x = 6 ± = 3 ± 2 5 The fully factored equation is (x + 2)(4x 1)(x ( ))(x ( 3 2 5)). The roots are 2, 1, ( ),( 3 2 5). 4 PTS: 1 DIF: Average REF: 15fcebf df-9c7d f0d2ea OBJ: Identifying All of the Real Roots of a Polynomial Equation NAT: NT.CCSS.MTH A.APR.3 LOC: MTH.C MTH.C TOP: 3-5 Finding Real Roots of Polynomial Equations 41. ANS: i 12i 1 = 1 + ( 12)i Rewrite as a + bi. = 1 ( 12)i Find a bi. = i Simplify. PTS: 1 DIF: Basic REF: a df-9c7d f0d2ea OBJ: Finding Complex Conjugates NAT: NT.CCSS.MTH N.CN.3 STA: DC.DCLS.MTH.05.AL2.AII.N.3 LOC: MTH.C TOP: 2-5 Complex Numbers and Roots KEY: complex numbers 19

23 42. ANS: A reflection across the x-axis and a vertical stretch by a factor of 7. Because 7 is negative, g is a reflection of f across the x-axis. Because 7 = 7, g is a vertical stretch of f by a factor of 7. PTS: 1 DIF: Basic REF: 1561f29a df-9c7d f0d2ea OBJ: Reflecting, Stretching, and Compressing Quadratic Functions NAT: NT.CCSS.MTH F.BF.3 STA: DC.DCLS.MTH.05.AL2.AII.P.11 LOC: MTH.C MTH.C MTH.C TOP: 2-1 Using Transformations to Graph Quadratic Functions KEY: quadratic graph 43. ANS: f(x) = x 2 4x 32 x = 8 or x = 4 Write the zeros as solutions for two equations. x 8 = 0 or x + 4 = 0 Rewrite each equation so that it is equal to 0. Apply the converse of the Zero-Product Property to write a 0 = (x 8)(x + 4) product that is equal to 0. 0 = x 2 4x 32 Multiply the binomials. f(x) = x 2 4x 32 Replace 0 with f(x) PTS: 1 DIF: Average REF: a df-9c7d f0d2ea OBJ: Using Zeros to Write Function Rules NAT: NT.CCSS.MTH A.APR.2 STA: DC.DCLS.MTH.05.AL2.AII.P.5 LOC: MTH.C MTH.C TOP: 2-3 Solving Quadratic Equations by Graphing and Factoring 20

24 44. ANS: Make a table. x f(x) = x 2 5x + 6 (x, f(x)) 2 f(x) = ( 2) 2 5( 2) + 6 ( 2, 20) 1 f(x) = ( 1) 2 5( 1) + 6 ( 1, 12) 0 f(x) = 0 2 5(0) + 6 (0, 6) 1 f(x) = 1 2 5(1) + 6 (1, 2) 2 f(x) = 2 2 5(2) + 6 (2, 0) Plot the ordered pairs and connect with a smooth curve. PTS: 1 DIF: Average REF: 155f903e df-9c7d f0d2ea OBJ: Graphing Quadratic Functions Using a Table STA: DC.DCLS.MTH.05.AL2.AII.P.5 DC.DCLS.MTH.05.AL2.AII.G.3 LOC: MTH.C TOP: 2-1 Using Transformations to Graph Quadratic Functions KEY: quadratic graph 21

25 45. ANS: 15i 15i 15 = 15i Ê i 2 7 ˆ Rewrite i Ë Á 15 as a product of i and power of i 2. = 15i( 1) 7 i 2 = 1. = 15i Simplify. PTS: 1 DIF: Average REF: 15a2795e df-9c7d f0d2ea OBJ: Evaluating Powers of i NAT: NT.CCSS.MTH N.CN.2 STA: DC.DCLS.MTH.05.AL2.AII.N.3 LOC: MTH.C TOP: 2-9 Operations with Complex Numbers 46. ANS: x = 6 ±2 5 x 2 12x + 36 = 20 ( x 6) 2 = 20 Factor the perfect square trinomial. x 6 = ± 20 Take the square root of both sides. x = 6 ± 20 Add 6 to each side. x = 6 ±2 5 Simplify. PTS: 1 DIF: Average REF: 15752c8a df-9c7d f0d2ea OBJ: Solving Equations by Using the Square Root Property NAT: NT.CCSS.MTH A.REI.4 STA: DC.DCLS.MTH.05.AL2.AII.P.6 DC.DCLS.MTH.05.AL2.AII.P.7 LOC: MTH.C ANS: i 6i( 4 3i) 24i 18i 2 Distribute. 24i 18( 1) Use i 2 = i Write in a + bi form. TOP: 2-4 Completing the Square PTS: 1 DIF: Basic REF: 15a2524e df-9c7d f0d2ea OBJ: Multiplying Complex Numbers NAT: NT.CCSS.MTH N.CN.2 STA: DC.DCLS.MTH.05.AL2.AII.N.3 LOC: MTH.C TOP: 2-9 Operations with Complex Numbers 22

26 48. ANS: 6x x (x 2) To divide, first write the dividend in standard form. Include missing terms with a coefficient of 0. 6x 3 + 0x 2 + 5x 8 Then write out in long division form, and divide. 6x x + 29 x 2 6x 3 + 0x 2 + 5x 8 (6x 3 12x 2 ) 12x 2 + 5x (12x 2 24x) 29x 8 (29x 58) 50 Write out the answer with the remainder to get 6x x (x 2). PTS: 1 DIF: Average REF: 15ea df-9c7d f0d2ea OBJ: Using Long Division to Divide Polynomials NAT: NT.CCSS.MTH A.APR.6 LOC: MTH.C TOP: 3-3 Dividing Polynomials 49. ANS: x = 2 + 9i or 2 9i x 2 4x + 85 = 0 Set f(x) = 0. x 2 4x = 85 Rewrite. x 2 4x + 4 = Ê b Add 2 Ë Á ˆ 2 to both sides of the equation. (x 2) 2 = 81 Factor. x 2 = ± 81 Take square roots. x = 2 ± 9i Simplify. PTS: 1 DIF: Average REF: 1580f df-9c7d f0d2ea OBJ: Finding Complex Zeros of Quadratic Functions NAT: NT.CCSS.MTH N.CN.7 NT.CCSS.MTH A.REI.4 STA: DC.DCLS.MTH.05.AL2.AII.P.5 LOC: MTH.C MTH.C TOP: 2-5 Complex Numbers and Roots KEY: complex numbers 23