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Chapter 2 Review 1. Topics to study include (but are not limited to) how to graph transformations and parabolas, to write the quadratic equation given a graph or information, to be able to perform operations on imaginary numbers, find the discriminant, calculate values of i, solve quadratic equations, and write equations of word problems modeled with quadratics. 2. Analyze the function . a. Is its graph concave up or concave down? b. What is the y-intercept? c. What is the axis of symmetry? d. What is the vertex? 3. Simplify each expression. Show all of your work. 5. A basketball player stands near the middle of the court and throws the ball toward the basket. The path of the ball is a parabola. The ball leaves the player’s hands at a height of 6 feet above the ground. The ball travels to a maximum height of 12 feet when it is a horizontal distance of 18 feet from the player. Write a function to represent the height of the ball in terms of its distance from the player. 6. Jeremy stands on the edge of a river and throws a rock into the water. The rock leaves his hand at a height 5 feet above the water. It moves upward and then down before splashing into the center of the river. The rock reaches the highest point along its path when it is a horizontal distance of 30 feet from his hand and 20 feet above the water. Write a function to represent the height of the rock in terms of its distance from Jeremy’s hand. 7. Simplify each expression. Show all of your work. a. a. b. b. c. c. 4. Analyze the graph. Describe whether each function could be used to model the graph. Explain your reasoning. 8. Determine the number of zeros and the type of zeros for each quadratic function. Justify your reasoning. a. b. a. b. Write a quadratic equation for the given information. 9. Given two x-intercepts ( point , 0) and (5, 0) and one 10. Given: exactly one x-intercept (2, 0) and y-intercept 11. Given: vertex and point Calculate each power of i. 12. 24. 13. 14. For each complex number, write its conjugate. 15. 16. 25. Calculate each quotient. 17. 18. 19. Graph 20. Graph Use the Quadratic Formula to solve an equation of the form for each function. without a calculator. without a calculator. Create a system of equations and use algebra to write a quadratic equation for each set of three points that lie on a parabola. 26. 27. Use the discriminant to determine whether each function has real or imaginary zeros. 21. 28. Use a graphing calculator to determine the quadratic equation for each set of three points that lie on a parabola. 22. 23. Write the function that represents each graph. 29. Chapter 2 Review Answer Section 1. ANS: . PTS: 1 2. ANS: a. concave up b. c. d. The vertex is . PTS: 1 REF: 2.1 NAT: A.SSE.1.a | A.SSE.2 | A.APR.1 | F.IF.4 | F.IF.9 TOP: Pre Test KEY: standard form of a quadratic function | factored form of a quadratic function | vertex form of a quadratic function | concavity of a parabola 3. ANS: a. b. c. PTS: 1 REF: 2.6 NAT: N.CN.1 | N.CN.2 | N.CN.3(+) | N.CN.8(+) TOP: Pre Test KEY: the imaginary number i | principal square root of a negative number | set of imaginary numbers | pure imaginary number | set of complex numbers | real part of a complex number | imaginary part of a complex number | complex conjugates | monomial | binomial | trinomial 4. ANS: a. The function could not model the graph because although its negative a-value means it is concave down, it has a positive y-intercept. b. The function could not model the graph because although its negative a-value means it is concave down, both of its x-intercepts are negative. PTS: 1 REF: 2.1 NAT: A.SSE.1.a | A.SSE.2 | A.APR.1 | F.IF.4 | F.IF.9 TOP: Mid Ch Test KEY: standard form of a quadratic function | factored form of a quadratic function | vertex form of a quadratic function | concavity of a parabola 5. ANS: PTS: 1 REF: 2.1 NAT: A.SSE.1.a | A.SSE.2 | A.APR.1 | F.IF.4 | F.IF.9 TOP: End Ch Test KEY: standard form of a quadratic function | factored form of a quadratic function | vertex form of a quadratic function | concavity of a parabola 6. ANS: PTS: 1 REF: 2.1 NAT: A.SSE.1.a | A.SSE.2 | A.APR.1 | F.IF.4 | F.IF.9 TOP: Mid Ch Test KEY: standard form of a quadratic function | factored form of a quadratic function | vertex form of a quadratic function | concavity of a parabola 7. ANS: a. b. c. PTS: 1 REF: 2.6 NAT: N.CN.1 | N.CN.2 | N.CN.3(+) | N.CN.8(+) TOP: End Ch Test KEY: the imaginary number i | principal square root of a negative number | set of imaginary numbers | pure imaginary number | set of complex numbers | real part of a complex number | imaginary part of a complex number | complex conjugates | monomial | binomial | trinomial 8. ANS: a. The function has 2 imaginary roots. Vertex is above the x-axis and opens up so it never intersects the x-axis. b. The function has 2 real zeros that are a double root. I know this because the function is given in vertex form with . PTS: 1 REF: 2.7 NAT: N.CN.7 | N.CN.8(+) | N.CN.9(+) TOP: End Ch Test KEY: imaginary roots | discriminant | imaginary zeros | Fundamental Theorem of Algebra | double root 9. ANS: PTS: 1 REF: 2.5 TOP: Skills Practice 10. ANS: NAT: A.CED.1 | F.IF.4 | F.BF.1.a PTS: 1 REF: 2.5 TOP: Skills Practice 11. ANS: NAT: A.CED.1 | F.IF.4 | F.BF.1.a PTS: 1 REF: 2.5 TOP: Skills Practice 12. ANS: NAT: A.CED.1 | F.IF.4 | F.BF.1.a PTS: 1 REF: 2.6 NAT: N.CN.1 | N.CN.2 | N.CN.3(+) | N.CN.8(+) TOP: Skills Practice KEY: the imaginary number i | principal square root of a negative number | set of imaginary numbers | pure imaginary number | set of complex numbers | real part of a complex number | imaginary part of a complex number | complex conjugates | monomial | binomial | trinomial 13. ANS: PTS: 1 REF: 2.6 NAT: N.CN.1 | N.CN.2 | N.CN.3(+) | N.CN.8(+) TOP: Skills Practice KEY: the imaginary number i | principal square root of a negative number | set of imaginary numbers | pure imaginary number | set of complex numbers | real part of a complex number | imaginary part of a complex number | complex conjugates | monomial | binomial | trinomial 14. ANS: PTS: 1 REF: 2.6 NAT: N.CN.1 | N.CN.2 | N.CN.3(+) | N.CN.8(+) TOP: Skills Practice KEY: the imaginary number i | principal square root of a negative number | set of imaginary numbers | pure imaginary number | set of complex numbers | real part of a complex number | imaginary part of a complex number | complex conjugates | monomial | binomial | trinomial 15. ANS: PTS: 1 REF: 2.6 NAT: N.CN.1 | N.CN.2 | N.CN.3(+) | N.CN.8(+) TOP: Skills Practice KEY: the imaginary number i | principal square root of a negative number | set of imaginary numbers | pure imaginary number | set of complex numbers | real part of a complex number | imaginary part of a complex number | complex conjugates | monomial | binomial | trinomial 16. ANS: PTS: 1 REF: 2.6 NAT: N.CN.1 | N.CN.2 | N.CN.3(+) | N.CN.8(+) TOP: Skills Practice KEY: the imaginary number i | principal square root of a negative number | set of imaginary numbers | pure imaginary number | set of complex numbers | real part of a complex number | imaginary part of a complex number | complex conjugates | monomial | binomial | trinomial 17. ANS: PTS: 1 REF: 2.6 NAT: N.CN.1 | N.CN.2 | N.CN.3(+) | N.CN.8(+) TOP: Skills Practice KEY: the imaginary number i | principal square root of a negative number | set of imaginary numbers | pure imaginary number | set of complex numbers | real part of a complex number | imaginary part of a complex number | complex conjugates | monomial | binomial | trinomial 18. ANS: . PTS: 1 REF: 2.6 NAT: N.CN.1 | N.CN.2 | N.CN.3(+) | N.CN.8(+) TOP: Skills Practice KEY: the imaginary number i | principal square root of a negative number | set of imaginary numbers | pure imaginary number | set of complex numbers | real part of a complex number | imaginary part of a complex number | complex conjugates | monomial | binomial | trinomial 19. ANS: I start by rewriting the function as . I plot the vertex at to get m(x) in the form because of m(x) is horizontally stretched by a factor of and . Compared to the graph of or 2 because , the graph . From the vertex, I go right 2 units and up 1 unit to plot the point and I go right 4 units and up 4 units to plot the point points across the line of symmetry and plot the corresponding points and points with a smooth curve to complete the graph. PTS: 1 TOP: Assignment 20. ANS: , where . I reflect these last 2 . I connect the REF: 2.4 NAT: F.IF.7.a | F.BF.3 KEY: horizontal dilation | horizontal stretching | horizontal compression I start by rewriting the function as . I plot the vertex at because of g(x) is horizontally compressed by a factor of to get g(x) in the form and . Compared to the graph of or because , where the graph . From the vertex, I go right 0.5 unit and up 1 unit to plot the point , I go right 1 unit and up 4 units to plot the point (5, 0), and I go right 1.5 units and up 9 units to plot the point (5.5, 5). I reflect these last 3 points across the line of symmetry and plot the corresponding points , (3, 0), and (2.5, 5). I connect the points with a smooth curve to complete the graph. PTS: 1 TOP: Assignment 21. ANS: Equation 1: REF: 2.4 NAT: F.IF.7.a | F.BF.3 KEY: horizontal dilation | horizontal stretching | horizontal compression Equation 2: Equation 3: Substitute equation 2 into equation 1 and solve for a. Substitute the value for a in terms of b and the value for c into equation 3 and solve for b. Substitute the values for b and c into equation 1 and solve for a. Substitute the values for a, b, and c into a quadratic equation in standard form. PTS: 1 REF: 2.5 TOP: Skills Practice 22. ANS: NAT: A.CED.1 | F.IF.4 | F.BF.1.a PTS: 1 REF: 2.5 TOP: Skills Practice 23. ANS: NAT: A.CED.1 | F.IF.4 | F.BF.1.a PTS: 1 REF: 2.5 TOP: Skills Practice 24. ANS: NAT: A.CED.1 | F.IF.4 | F.BF.1.a PTS: 1 REF: 2.3 NAT: F.IF.7.a | F.BF.3 TOP: Skills Practice KEY: vertical dilation | vertical stretching | vertical compression | reflection | line of reflection 25. ANS: PTS: 1 REF: 2.3 NAT: F.IF.7.a | F.BF.3 TOP: Skills Practice KEY: vertical dilation | vertical stretching | vertical compression | reflection | line of reflection 26. ANS: PTS: 1 REF: 2.7 NAT: N.CN.7 | N.CN.8(+) | N.CN.9(+) TOP: Skills Practice KEY: imaginary roots | discriminant | imaginary zeros | Fundamental Theorem of Algebra | double root 27. ANS: PTS: 1 REF: 2.7 NAT: N.CN.7 | N.CN.8(+) | N.CN.9(+) TOP: Skills Practice KEY: imaginary roots | discriminant | imaginary zeros | Fundamental Theorem of Algebra | double root 28. ANS: The discriminant is zero, so the function has real zeros (double roots). PTS: 1 REF: 2.7 NAT: N.CN.7 | N.CN.8(+) | N.CN.9(+) TOP: Skills Practice KEY: imaginary roots | discriminant | imaginary zeros | Fundamental Theorem of Algebra | double root 29. ANS: The discriminant is positive, so the function has real zeros. PTS: 1 REF: 2.7 NAT: N.CN.7 | N.CN.8(+) | N.CN.9(+) TOP: Skills Practice KEY: imaginary roots | discriminant | imaginary zeros | Fundamental Theorem of Algebra | double root