Name Period Date QUAD2.1 QUAD2.2 QUAD2.3 The Square Root Property Solve quadratic equations using the square root property Understand that if a quadratic function is set equal to zero, then the result is a quadratic equation whose roots are equal to the x-intercepts of the function. The Zero Product Property Solve quadratic equations using the zero product property Understand that if a quadratic function is set equal to zero, then the result is a quadratic equation whose roots are equal to the x-intercepts of the function. The Rabbit Pen Model a situation using mathematics Use quadratic functions and equations to solve a problem. QUAD2 STUDENT PAGES QUADRATIC FUNCTIONS AND EQUATIONS Student Pages for Packet 2: Solving Quadratic Equations 1 QUAD2.4 Vocabulary, Skill Builders, and Review 22 1 10 16 QUAD2 SP
Word difference of two squares maximum point minimum point perfect square trinomial quadratic equation quadratic regression square root property zero product property WORD BANK (QUAD2) Definition Example or Picture QUAD2 SP0
2.1 The Square Root Property Ready (Summary) THE SQUARE ROOT PROPERTY We will learn to use the square root property to solve certain quadratic equations. We will connect this procedure to some special factoring cases and to the graph of quadratic functions. Go (Warmup) Set (Goals) Solve quadratic equations using the square root property Understand that if a quadratic function is set equal to zero, then the result is a quadratic equation whose roots are equal to the x-intercepts of the function. Solve each equation. 1. x 16 + 2x 17 = 0 2. -4 + 2x + 8 = 3(x + 3) (x + 5) 3. x 9 = x + 9 4. 1 3 x = 2x+ 2 2 4 QUAD2 SP1
2.1 The Square Root Property RADICALS REVISITED A square root of a number n is a number whose square is equal to n, that is, a solution of the equation x 2 = n. The positive square root of a number n, written n, is the positive number whose square is n. The negative square root of a number n, written - n, is the negative number whose square is n. Estimate each expression as a decimal to the nearest tenth. 1. 30 2. 5 + 30 3. 5 30 4. 6 5. 8+ 6 6. 8 6 Simplify each expression. A (square root) radical expression is considered simplified if: 1. There are no perfect squares under the radical. 2. There are no fractions under the radical. 3. There are no radicals in the denominator. 7. 25 8. ± 25 9. 25 10. 2 (25) 11. 36 + 64 12. 36 + 64 13. 20 14. 16. 9 4 17. 4+ 8 3 4 15. 2 3 18. 4 + 8 2 QUAD2 SP2
2.1 The Square Root Property Factor. Look for patterns. 1. FACTORING: SPECIAL CASES REVISITED a. x 2 + 10x + 25 = b. x 2 14x + 49 = c. x 2 + 2x + 1 = d. x 2 6x + 9 = Describe the pattern and write it in a general form. This pattern is called a Factor the following using any method. Put a star happy face by the difference of two squares. 2. a. x 2 25 = b. x 2 49 = c. x 2 1 = d. x 2 9 = Describe the pattern and write it in a general form. This pattern is called the 1. x 2 + 10x + 25 2. x 2 16 3. x 2 + 13x + 36 4. x 2 8x + 16 5. x 2 81 6. x 2 + 6x 16 by the perfect square trinomials. Put a QUAD2 SP3
2.1 The Square Root Property EXPLORING THE SQUARE ROOT PROPERTY 1. Fill in the t-table and draw the graph for this quadratic function. x y = x 2 9 4 3 2 1 0-1 -2-3 -4 2. Write the x-intercepts for the graph as coordinate pairs. 3. If y = x 2 9 and y = 0, then 0 = x 2 9 (which can be written: x 2 = 9). What value(s) of x make this equation true? ( ) 2 = 9 = ( ) 2 = 9 = 4. How are the zeros (sometimes called solutions) of quadratic equation and the graph of the quadratic function related? y x QUAD2 SP4
2.1 The Square Root Property EXPLORING THE SQUARE ROOT PROPERTY (continued) 1. If x 2 = 9, then x = or x = (sometimes written x = ) 2. If x 2 = 49, then x = or x = (sometimes written x = ) 3. If x 2 = a 2, then x = or x = (sometimes written x = ) The square root property states that if x 2 = a 2, then x = ± a 4. Build the equation x 2 + 10x + 25 = 100 using algebra pieces. Arrange both sides of the equation into squares. Then use the square root property to solve this quadratic equation: Write the equation. x 2 + 10x + 25 = 100 Write the quadratic polynomial as a perfect square. Apply the square root property. Solve for x. Check both solutions in the original equation. ( ) 2 = 100 ( ) = ± 10 = 10 or = -10 x = or x = For x =, check: ( ) 2 + 10( ) + 25 = 100 Remember: two solutions ( ± ) For x =, check: ( ) 2 + 10( ) + 25 = 100 QUAD2 SP5
2.1 The Square Root Property EQUATION SOLVING PRACTICE 1 1. Restate the square root property in your own words. Solve. Check your solution(s) in the original equation. 2. x 2 = 9 3. (x + 3) 2 = 9 4. (x 10) 2 = 81 5. x 2 + 6x + 9 = 64 6. 3x + 1 = 121 7. x 2 + 4x + 4 = 36 QUAD2 SP6
2.1 The Square Root Property MORE ABOUT THE SQUARE ROOT PROPERTY The square root property applies when constant expression is not a perfect square. 1. If x 2 = 5, then x = or x = (sometimes written x = ) 2. If x 2 = 20, then x = or x = (sometimes written x = ) 3. If x 2 = a, then x = or x = (sometimes written x = ) Alternate Statement of the Square Root Property The square root property states that if x 2 = a, then x = ± a. a. Find solutions in simplest radical form. 2 ( x + 2) = 12 x + 2 = ± 12 x = 2 ± 12 x = 2± 43 x = 2± 4 3 x = 2± 2 3 x = 2 + 2 3 or x = 2 2 3 2 Example: Solve ( x + 2) = 12 b. Estimate solutions as decimals. We know that x = 2 ± 12 9 < 12 < 16 3 < 12 < 4 An estimate for 12 is 3.5 If x = 2 ± 3.5, then x 5.5 or x 1.5 Solve. Write solutions in simplest radical form and estimate solutions as decimals. 4. x 2 = 7 5. (x + 3) 2 = 28 QUAD2 SP7
2.1 The Square Root Property EQUATION SOLVING PRACTICE 2 Solve. Check your solution(s) in the original equation. 1. (x + 4) 2 = 16 2. x 2-11 = -11 3. (x 7) 2 = 50 4. x 2 + 2x + 3 = 2 5. x 2-7 = 0 6. -3(x 7) = 2(x + 6) + x + 9 7. (x + 3) 2 = 1 4 8. x 2 = -9 9. Describe the kind of equations that can be solved using the square root property. QUAD2 SP8
2.1 The Square Root Property FIND THE MISTAKE Jenna tried to solve this equation and explain what she did, but she thinks she made a mistake. Put an X next to step that is incorrect. Jenna s Work Explanations 1. x 2 original equation + 6x + 7 = 18 +2 +2 2. addition property of equality 3. x 2 + 6x + 9 = 20 arithmetic 4. (x + 3) 2 = 20 factor left side 5. x + 3 ( ) 2 = 20 Take square root of both sides 6. (x + 3) = 20 arithmetic 7. x + 3 = 2 5 arithmetic 8. arithmetic x + 3 = 2 5 9. 3 3 multiplication property of equality 10. x = -3 + 2 5 arithmetic 11. x 5.8 arithmetic Now rework the problem above, beginning from the step where the first error occurred. Your corrected work Explanations QUAD2 SP9
2.2 The Zero Product Property Ready (Summary) THE ZERO PRODUCT PROPERTY We will learn to use the zero product property to solve certain quadratic equations by factoring. We will connect this procedure to the graphs of quadratic functions. Go (Warmup) Set (Goals) Solve quadratic equations using the zero product property Understand that if a quadratic function is set equal to zero, then the result is a quadratic equation whose roots are equal to the x-intercepts of the function. Solve each equation. 1. x 2 = 144 2. (x 5) 2 = 4 3. x 2 + 8x + 16 = 25 4. Factor x 2 + 2x 3 using any method. QUAD2 SP10
2.2 The Zero Product Property EXPLORING THE ZERO PRODUCT PROPERTY 1. Fill in the t-table and draw the graph for this quadratic function. x y = x 2 + 2x 3 4 3 2 1 0-1 -2-3 -4 2. Write the x-intercepts for the graph as coordinate pairs. 3. If y = x 2 + 2x 3 and y = 0, then 0 = x 2 + 2x 3 (which can be written x 2 + 2x 3 = 0). Write the left side of this quadratic equation in factored form. ( )( ) = 0 4. What value(s) of x make this equation true? Why? 5. How are the solutions (also known as zeros) of a quadratic equation and the graph of the quadratic function related? y x QUAD2 SP11
2.2 The Zero Product Property EXPLORING THE ZERO PRODUCT PROPERTY (continued) 6. 5 = 0 7. 9 = 0 8. 0 = 0 The zero product property states that if ab = 0, then a = 0 or b = 0. 9. If ab = 0, then what do we know about a and b? 10. Use the zero product property to solve this quadratic equation: x 2 x = 2 Write the equation. x 2 x = 2 Rewrite so that the quadratic polynomial is equal to zero. = 0 Factor the polynomial, ( )( ) = 0 Apply zero product property. = 0 or = 0 Solve for x. x = or x = Check both solutions in the original equation. For x =, check: ( ) 2 ( )= 2 For x =, check: ( ) 2 ( )= 2 QUAD2 SP12
2.2 The Zero Product Property EQUATION SOLVING PRACTICE 1. Restate the zero product property in your own words. Solve using the zero product property. 2. (x 5)(x + 4) = 0 3. (x + 1)(x + 6) = 0 4. x 2 + 9x + 14 = 0 5. x 2 + 2x 15 = 0 6. x 2 9x = -20 7. x 2 = 8x + 48 QUAD2 SP13
2.2 The Zero Product Property EQUATION SOLVING PRACTICE (continued) Solve each equation. Placing them in factored form first may be helpful. 8. x 2 + 11x + 24 = 0 9. x 2 25 = 0 10. x 2 10x = -21 11. x 2 + 9x = 22 12. x 2 8x = 0 13. x = 3x + 40 14. Describe the kind of equations that can be solved using the zero product property. QUAD2 SP14
2.2 The Zero Product Property FIND THE MISTAKES 1. Matt tried to solve the equation below. Circle the step where he made an error. Then rework the problem below, beginning from the first step in which an error is found. 1. 2. 3. 4. 5. Matt s Work x 2 + 8x + 12 = -3 (x + 2)(x + 6) = -3 x + 2 = -3 or x + 6 = -3-2 -2 or -6-6 x = -5 or x = -9 Explain Matt s error. Do you promise to never do this? Corrected work 2. Blakely tried to solve the equation below. Circle the step where he made an error. Then rework the problem below, beginning from the first step in which an error is found. Blakely s Work 1. x 2 + x - 12 = 0 2. (x + 4)(x 3) = 0 3. x = 4 or x = - 3 Explain Blakely s error. Do you promise to never do this? Corrected work QUAD2 SP15
2.3 The Rabbit Pen Ready (Summary) We will use our knowledge of length, area, and quadratic functions and equations to find the largest area possible, given a fixed perimeter. THE RABBIT PEN Go (Warmup) Set (Goals) Model a situation using mathematics. Solve a problem involving maximization. Use quadratic functions and equations to solve a problem. Find the missing measurements for each rectangle. (Note: Diagrams are not to scale.) 1. A = 4. 7 cm 6 yd A = 36 yd 2 7 cm P = P = 2. A = 5. 9 in A = 2 mm 3 in P = P = 16 mm 3. A = 6. P = 6 ft 5 cm x ft A = 5x cm 2 P = QUAD2 SP16
2.3 The Rabbit Pen MAKING A RABBIT PEN 1. You have 40 feet of wire mesh to make a rectangular rabbit pen. Sketch several possible rectangles that you could create. Label the length and width of each. (Let L be the length of the horizontal side and W be the length of the vertical side.) 2. For each rectangle, what do you notice about L+W? 3. What is the maximum area that you found for a rabbit pen? QUAD2 SP17
2.3 The Rabbit Pen THE RABBIT PEN: LENGTHS AND AREAS 1. Use the pictures and data you created on the previous page to complete the table at the right. Recall that you have 40 feet of wire mesh. Complete the table in any order. 2. Write a formula to find area of a rectangle: A = 3. Write two different formulas to find perimeter of a rectangle: P = P = 4. What is the perimeter of each rabbit pen? 5. What must be the sum of one length and one width? L + W =. Length in ft. (L) 0 2 4 Width in ft. (W) Area in sq. ft (A) 6 14 84 6. What is strange about the first row of the table? In other words, what seems problematic about the area measurement associated with L = 0? 7. Why can there not be a rectangle with L = 22? 8. What is the maximum area that you found in the table? 9. What is special or different about this rectangle? 8 10 12 14 16 18 20 L QUAD2 SP18
2.3 The Rabbit Pen RABBIT PENS: WIDTH VS. LENGTH 1. Graph the data for width vs. length from the previous table. Scale the axes appropriately. 2. Find an equation for the graph using your knowledge of linear functions. 3. Find a function for the graph using known facts about the rabbit pen. a. Write a general formula for the perimeter of a rectangle. b. Substitute the known value for perimeter for the rabbit pen. c. Solve for W in terms of L. 4. Do the two equations agree? Explain. 5. Interpret your graph. Width Length a. What is the domain? What is the range? b. Why are they restricted? c. What is the x-intercept? What is the y-intercept? d. What do these points represent in the context of the rabbit pen problem? QUAD2 SP19
2.3 The Rabbit Pen RABBIT PENS: AREA VS. LENGTH 1. Graph the data for area vs. length from the previous table. Scale the axes appropriately. 2. Find a function for the graph using known facts about the rabbit pen. a. First recall the formula for W in terms of L from the previous page. b. Write the general formula for the area of a rectangle. c. Make a substitution for W so that your function for area is in terms of L. (This expression will be in factored form.) d. Write this function as a sum of terms in standard form. Area Length 3. Find a function for the rabbit pen data using a graphing calculator or a web-based quadratic regression application. 4. Do the two functions for area vs. length agree? Explain. QUAD2 SP20
2.3 The Rabbit Pen RABBIT PENS: AREA VS. LENGTH (continued) 5. Interpret your graph. a. What is the domain? What is the range? Why are they restricted? b. What are the x-intercepts? What is the y-intercept? What do these points represent in the context of the rabbit pen problem? c. Is there a maximum or a minimum? If so, what is it, and what does it represent in the context of the rabbit pen problem. d. What (if any) of these points is the vertex? 6. Explore the quadratic equation obtained by setting the quadratic function equal to zero. a. Write this quadratic equation in factored form. b. Find the solutions to this quadratic equation. c. Interpret the solutions to the quadratic equation in the context of the rabbit pen problem and the area vs. length graph. QUAD2 SP21
2.4 Vocabulary, Skill Builders, and Review FOCUS ON VOCABULARY (QUAD2) Match each phrase with its explanation or example. 1. 2. 3. 4. 5. 6. 7. 8. 9. 1 0. square root property a. An equation of the form 2 ax + bx + c = 0 zero product property b. A coordinate pair that has the greatest y-value for a function difference of two squares c. 2 2 if x = a then x = ± a. perfect square trinomial d. 2 x 9 = ( x+ 3)( x 3) maximum point of a function e. A process by which an equation for a parabola that best fits data is found minimum point of a function f. another name for x-intercept quadratic equation g. A coordinate pair that has the least y-value for a function. quadratic regression h. if ab = 0, then a = 0 or b = 0. root of an equation i. x 6x+ 9 = ( x 3) 2 2 zeroes of a function j. a solution of an equation QUAD2 SP22
2.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 1 Simplify each expression. If it is already in simplest form, circle the expression. 1. 9 2. 5. 3 6. x + 3 9. 11 25 13. 7 5 4 10. 2 3 3. 3 3 4. 11 25 14. 5 Estimate each expression to the nearest tenth. 7. 11. 16 25 11 5 15. -2 + 17. 20 18. 5 + 20 19. 5 20 9 4 8. 12. ( 3) 16 25 25 11 16. -2 20. 2 9 4 5 20 2 QUAD2 SP23
2.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 2 Fill in the t-table and draw the graph for each equation. 1. x y = x 2 + 2 y = x 2 2 2 1 0-1 -2 2. How are the parabolas on the right similar? How are they different? It s time to review your skills in scientific notation from earlier this year. Your future science teacher is going to appreciate your effort here! Make sure your answer is in proper scientific notation. An example is done for you. Written in Proper Scientific Notation Written without Scientific Notation y Is this number really small or really large? Ex: (2 10 50 ) (4 10 40 ) 2.4 10 14 240,000,000,000,000 Really Large 3. (5 107 )(3 10 6 ) 4. (2 10 17 )(4.1 10 6 ) 5. (2.3 10 2 )(8 10 5 ) 6. (2 10 50 ) (4 10 40 ) x QUAD2 SP24
2.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 3 Fill in the t-table and draw the graph for each equation. 1. 2. x y = 3x 2 y = 3x 2 + 4 2 1 0-1 -2 3. Multiply (x 6)(x + 9) Simplify. 4. 2(x 2 x) 5(x 2) 5. 3(x + 1)(x + 12) 4x(x 7) Factor. 6. x 2 11x + 18 7. x 2 x 42 8. x 2 100 9. x 2 + 6 x + 5 5 y x QUAD2 SP25
2.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 4 Fill in the t-table and draw the graph for each equation. 1. x y = 2x 2 y = 1 2 x2 2 1 0-1 -2 2. How are the parabolas on the right similar? How are they different? Solve for x. 3. (x 1) 2 = 81 4. x 2 2x 15 = 0 5. x 2 + 16x + 64 = 9 6. x 2 = -6x + 40 7. x 2 18 = -x 6 8. x 2 + 4x = 4(x + 4) QUAD2 SP26 y x
2.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 5 Which of the following input-output tables illustrate functions? If NO, state why not. 1. 2. x (input) -3-2 -1 0 1 2 3 y (output) 12 7 4 3 4 7 12 x (input) 3 2 1 0 1 2 3 y (output) 9 4 1 0-1 -4-9 Which of the following ordered pairs illustrate functions? If NO, state why not. 3. (1,1), (2,2), (3,3), (4,4) 4. (10,8), (-10,-8), (8,10), (-8,-10) 5. (0,10), (0.5,10), (1,10), (1.5,10) Which of the following mapping diagrams illustrate functions? If NO, state why not. 6. 7. a a d b b 8. c -1-3 -5 c -13-1 9. -7 0 2 4 5 5 7 6-2 -9 11 8 10 12 6 QUAD2 SP27
2.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 6 On each graph below, the x-axis is the horizontal axis and the y-axis is the vertical axis. Which of the following graphs illustrate functions of x? If it is not a function, state why not. If it is a function, state the domain and range. 1. 2. 3. 4. 5. Sketch a nonlinear graph that represents a function 6. Sketch a linear graph that does NOT represent a function QUAD2 SP28
2.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 7 1. Graph the lines y 1 = 1 2 x 2 and y 2 = 2x + 4 using two different colored pencils. 2. Find the intercepts for each function. y 1 = 1 2 x 2 y 2 = 2x + 4 x-intercept (, ) (, ) y-intercept (, ) (, ) 3. Graph a parabola by multiplying y-coordinates from the lines. Use a third color. y 4. Find an equation for the parabola. y 3 = y 1 y 2 = 5. Identify the following coordinates for the parabola: a. x-intercept(s) b. y-intercept(s) c. vertex 6. Is the vertex a minimum or a maximum point? What are its coordinates? 7. What is the domain of the parabola s function? 8. What is the range of the parabola s function? x QUAD2 SP29
2.4 Vocabulary, Skill Builders, and Review Write the following numbers without exponents. SKILL BUILDER 8 1. 8 2 2. (-8) 2 3. -8 2 Evaluate the following expressions for a = 2, b = -5, and c = -3. 4. -b 5. a - c 6. a c b 7. b 2 8. - 4ac 9. b 2 4ac 10. 12 ± 20b 4 11. 12 ± 20b 4 13. Joey and Maria are trying to solve x 2 9 = 0. Joey s work x 2 9 = 0 x 2 = 9 x = ±3 What property did Joey use in his work to solve the equation? What property did Maria use in her work to solve the equation? 12. b2 ± 20b 4 Maria s work x 2 9 = 0 (x 3)(x + 3) = 0 x 3 = 0 x + 3 = 0 x = 3 x = -3 Whose method would you use to solve x 2 16 = 0? Why? QUAD2 SP30
2.4 Vocabulary, Skill Builders, and Review TEST PREPARATION (QUAD2) Show your work on a separate sheet of paper and choose the best answer. 1. Circle the letter that represents the complete solution(s) to: (x + 3) 2 = 17 A. x = 3 + 17 B. x = ± 14 C. 3 ± 17 2. Which quadratic function has a graph with x-intercepts at x = -3 and x = 5? A. f(x) = x 2 + 2x + 15 B. f(x) = x 2 + 8x + 15 C. f(x) = x 2-2x - 15 D. f(x) = x 2-8x + 15 3. Circle the statement(s) that are NOT true about the function f(x) = x 2 + 7x 8. D. None of these A. The factored form is f(x) = (x + 8)(x 1). B. The graph of f(x) has an x-intercept when x = -8. C. The graph of f(x) has an x-intercept when x = 8. D. The graph of f(x) has an x-intercept when x = 1. 4. Circle all of the following expressions that are perfect trinomials? A. x 2 + 4x + 8 B. x 2 + 8x + 16 C. x 2 + 6x + 9 D. x 2 12x + 144 5. Which of the following expressions represents the zero product property rule? A. a + 0 = a B. a is undefined. 0 C. If ab = 0, then a = 0 and/or b = 0. D. If ab = 6, then a = 2 and b = 3. 6. Farmer Wayne has 100 feet of fencing to use for his pigpen. What size and shape should he make his pen to maximize the area? A. A rectangle with a length of 30 feet and B. A square with sides of 50 feet. a width of 20 feet. C. A square with sides of 25 feet. D. A rectangle with a length of 70 feet and a width of 30 feet. QUAD2 SP31
2.4 Vocabulary, Skill Builders, and Review KNOWLEDGE CHECK (QUAD2) Show your work on a separate sheet of paper and write your answers on this page. QUAD 2.1 The Square Root Property 1. Solve using the square root property: x 2 = 25 2. Solve using the square root property: (x - 3) 2 = 50 3. Solve using the square root property: x 2 + 10x + 25 = 100 QUAD 2.2 The Zero Product Property 4. Solve using the zero product property: (x + 3)(x 4) = 0 5. Solve using the zero product property: x 2 + 10x = 11 QUAD 2.3 The Rabbit Pen 6. Given a fixed perimeter, what kind of quadrilateral has a maximum area? QUAD2 SP32
HOME-SCHOOL CONNECTION (QUAD2) Here are some questions to review with your young mathematician. Solve the following equations using the square root property: 1. (x + 7) 2 = 81 2. x 2 6x + 9 = 49 Solve the following equations using the zero product property: 3. (x + 7)(x 8) = 0 4. x 2 + 2x = 8 5. At what values of x does the graph of f(x) = x 2 + 3x 4 cross the x-axis? Parent (or Guardian) signature QUAD2 SP33
COMMON CORE STATE STANDARDS FOR MATHEMATICS A-SSE-1b* Interpret expressions that represent a quantity in terms of its context: Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P. A-SSE-3a* Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression: Factor a quadratic expression to reveal the zeros of the function it defines. A-CED-2* Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. A-CED-3* Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. A-CED-4* Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm s law V = IR to highlight resistance R. A-REI-1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. A-REI-4b Solve quadratic equations in one variable: Solve quadratic equations by inspection (e.g., for x 2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. F-IF-4* For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. F-IF-5* Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. F-IF-7a* Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases: Graph linear and quadratic functions and show intercepts, maxima, and minima. F-IF-8a Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function: Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. MP1 MP2 MP3 MP4 MP5 MP6 MP7 MP8 STANDARDS FOR MATHEMATICAL PRACTICE Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. QUAD2 SP34