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Name: Hour: Mrs. Gorsline Integrated Math 2 Unit 2 Notes: Chapter 6 6-1 Homework Notes 6-1 Quadratic Expressions, Rectangles, and Squares (p. 346) ________________________________ is a quadratic expression in standard form (notice no =, so can’t be an equation) ________________________________ is a quadratic equation ________________________________ is a quadratic function (f:x-> is the same as f(x)) The product of any two linear expressions _________ and _________ is a _______________________ expression. (Remember FOIL?) Example: Simplify (2 x 3)(4 x 5) When a linear expression is multiplied by itself, the result is a __________________________. Simplifying a binomial squared as a quadratic expression is called _____________________________________________. Example: Expand ( x 7) 2 Complete p. 349 #5,13,15 Name: Hour: Mrs. Gorsline Integrated Math 2 Unit 2 Notes: Chapter 6 6-1 In Class Notes Real life problems can be solved using quadratic equations as well. Example: Suppose a rectangular swimming pool 50 m by 20 m is to be built with a walkway around it. If the walkway is w meters wide, write the total area of the pool and walkway in standard form. FOIL is not the only way to multiply binomials. You can also think of them as pieces of a side of a ___________ Example: Use the Square technique to multiply the following: (2 x 3)(4 x 5) ( x y)2 Binomial Square theorem (the shortcut method for ( x y ) ) 2 Example: Expand k (3m )2 4 Name: Hour: Mrs. Gorsline Integrated Math 2 Unit 2 Notes: Chapter 6 6-2 Homework Notes Geometrically, the _______________________ of a number n, written ______, is the __________________ from n to ____ on the number line. Algebraically, the ________________________ of a number can be defined piecewise as Example 1: Solve for x: x 2 5.3 Complete p. 355 #14 – 18 Name: Hour: Mrs. Gorsline Integrated Math 2 Unit 2 Notes: Chapter 6 6-2 In-Class Notes f ( x) x x is called the __________________________________ function. Graph this below. x You can use absolute value symbols when evaluating all solutions to ________________________. For all real numbers x, Example: Solve the following x 2 81 x 2 40 Example: A square and a circle have the same area. The square has side 10. What is the radius of the circle? Review: What is the difference between a rational number and an irrational number? Name: Hour: Mrs. Gorsline Integrated Math 2 Unit 2 Notes: Chapter 6 6-3 Homework Notes 6-3: Graph Translation Theorem In a relation described by a sentence in x and y, replacing the ____ with ____________ translates the graph _______________________ by _____ units. Replacing the ____ with ____________ translates the graph _______________________ by _____ units. For the quadratic y ax , 2 Translates the parabola ______________________ by _____ units and _________________________ by _____ units. We often use ___________ to represent this translation. Complete p. 361-632 #1,5,6,10-13 6-3 In-Class Notes Remember from yesterday’ calculator activity, a large a value makes the graph _________________, a small a value makes the graph _____________________, a negative a makes the graph ______________________ and a positive a makes the graph _______________________________. Name: Hour: Mrs. Gorsline Integrated Math 2 Unit 2 Notes: Chapter 6 Example 1: Sketch the graph of y 7 3( x 6) by first graphing y 3 x 2 Since the vertex of the graph of y ax 2 2 is at _______ , then the vertex of the graph _______________________ is at __________. Because of this, ________________________ is called the _______________________ of an equation for a parabola. The line with the equation ___________ is called the ________________________ __________________________. If the graph opens ________, then the vertex is the ________________________ y-value of the graph. If the graph opens ________, then the vertex is the ________________________ y-value of the graph. 1 2 Example: Sketch the graph of y ( x 3) and give the equation for the axis of symmetry. 2 Name: Hour: Mrs. Gorsline Integrated Math 2 Unit 2 Notes: Chapter 6 Be aware that _________ and _________ can replace the x and y in any function to translate it. Example: Sketch the graph of y 4 x 3 6-5 Homework Notes Finish the attached worksheets that go along with the TI-Nspire file “Completing the Square”. Use the space below to complete any problems that you don’t have room to complete on the worksheet. Complete p. 374 #2-5 Name: Hour: Mrs. Gorsline Integrated Math 2 Unit 2 Notes: Chapter 6 6-5 In-Class Notes You have now seen two forms for an equation of a parabola: Standard Form Vertex Form Because they are both useful in its own way, we need to know how to convert from one to the other. Converting Vertex Form to Standard Form will be learned in 6-4. Converting from Standard Form to Vertex form is a bit trickier. It requires you to ______________________ ___________________. In last night’s notes, you learned how to do this in order to solve for the ________ _______________________________. Today you will learn how to use this to get from standard form to ________________________ (when the equation = y). In order to complete the square, you need to be able create a ______________________________________. This can be done in one of two ways: Example: Use the reverse box method to make a perfect square trinomial for x 10 x 2 Example: Use the reverse binomial square theorem to make a perfect square trinomial for x 2 10 x . Name: Hour: Mrs. Gorsline Integrated Math 2 Unit 2 Notes: Chapter 6 Example: Now we will convert the equation y x 2 10 x 8 into vertex form. Step 1:_______________________________________________________________________________ Step 2:_______________________________________________________________________________ Step 3:_______________________________________________________________________________ Step 4:_______________________________________________________________________________ Step 5:_______________________________________________________________________________ Example: Convert the equation Step 1: Step 2: Step 3: Step 4: Step 5: y 3x 2 8x 9 into vertex form. Name: Hour: Mrs. Gorsline Integrated Math 2 Unit 2 Notes: Chapter 6 6-4 Homework Notes 6-4: Graphing y ax 2 bx c Example: Convert y 7 3( x 6) to standard form. Be sure to follow Order of Operations! 2 Graph the two equations to make sure that you get the same graph. Do this on both your TI-Nspire and your TI-84 so that you practice graphing on both calculators. Complete p. 367 #1-3 6-4 In-Class Notes 6-4: Graphing y ax bx c 2 Example: Show that y 2( x 3) 8 and 2 y 2 x 2 12 x 10 are equivalent. Do this first algebraically then by graphing. Make notes if needed so that you know how to do this both on the Nspire as well as the TI-84. Name: Hour: Mrs. Gorsline Integrated Math 2 Unit 2 Notes: Chapter 6 Theorem: The graph of the equation ___________________ is a parabola _________________ to the graph of ________ The graph of every quadratic function is a ______________________ with a y-intercept at ____. The __________________ of all quadratic functions has a domain of _______________________________. However, the _______________ of a quadratic function is different for every graph and depends on two things 1. Does the graph have a maximum (when ________) or does the graph have a minimum (when _________). 2. Where is that max or min (the _____________________) The __________________ is easy to find if your quadratic is in ____________________________. In standard form, you must use the process below to find the vertex. a. Use the equation ___________ to find the axis of symmetry, which is also the _______________ of the vertex. b. Plug x into the quadratic equation to find the ______________ of the vertex. Example: Suppose the following quadratic equation represents the vertical height h of a thrown baseball after t seconds. h 16t 2 44t 5 a. Find the vertex of this quadratic. b. Doe this function have a maximum or a minimum? Explain how you know. c. Find h when t = 0, 1, 2, and 3 d. Explain what each pair (t, h) tells you about the height of the ball. Name: Mrs. Gorsline Integrated Math 2 Hour: Unit 2 Notes: Chapter 6 e. Graph this function. Find the domain and range of this function (be careful because this is a real-life problem) In the previous problem, how high does the ball get? Approximately when does it hit the ground? This example is a special case of a general formula for the ____________ h of an object at time t with an initial _________________________________ v0 and initial ____________ h0 that was discovered by ______________. That formula is Where g is a constant measuring the _____________________________________. g is about _________ or _________. Be careful that you understand that this does not describe the _____________ of the ball, it only describes how the ____________ of the ball changes over _____________. If you threw a ball straight up in the air, the graph would still look like a _______________________. Complete p. 367 #2,3,5-14 on a separate sheet. Name: Hour: Mrs. Gorsline Integrated Math 2 Unit 2 Notes: Chapter 6 6-7 Homework Notes 6-7: The quadratic formula Any quadratic equation equaling zero can be solved using _________________________________. However, there are times when the arithmetic for this is quite complicated. When this happens, you can use the _______________________________. Quadratic Formula Theorem If ax 2 bx c 0 and a 0 , then We will be completing the proof for this in class tomorrow. Example: Solve 3x 2 11x 4 0 using the quadratic formula. Complete p. 385 #6 – 10 Name: Hour: Mrs. Gorsline Integrated Math 2 Unit 2 Notes: Chapter 6 6-7 In-Class Notes FYI: You will have a 5-point quiz on Monday that deals with the quadratic formula with no calculator. Proof of the Quadratic Formula: Given: the equation Steps ax 2 bx c 0 , where a 0 Justifications Name: Hour: Mrs. Gorsline Integrated Math 2 Unit 2 Notes: Chapter 6 Example: Pop Fligh, the famous baseball player, hit a pitch that followed the equation h( x) .005 x 2 2 x 3.5 . Find out when the ball was exactly 8 feet high. Remember that the quadratic formula can only be applied when an equation is in the ________________________ of a quadratic equation that is equal to _____. Example: The 3-4-5 right triangle has sides with consecutive integers. Are there any others with the same property? Complete p. 385-386, #4,5,17-20 on a separate sheet. Name: Hour: Mrs. Gorsline Integrated Math 2 Unit 2 Notes: Chapter 6 6-8 & 6-9 Homework Notes 6-8 and 6-9: Imaginary and Complex Numbers Example: Solve the equation t 400 2 What is the problem with this? Definition: When k 0 , the two solutions to x k are denoted _______________ and ___________. 2 Numbers that have negatives under a square root are called ______________________. Definition: __________ Example: Solve t 2 400 using the letter i. Simplify the following: a) (2i)(5i) b) 9 25 Complete p. 391 #3,5 and p. 397 #2,4,6 c) 27 3 Name: Hour: Mrs. Gorsline Integrated Math 2 Unit 2 Notes: Chapter 6 6-8 & 6-9 In-Class Notes 6-8 and 6-9: Imaginary and Complex Numbers ____________ is called the _____________________ number. When you take ___ to different powers, a pattern emerges: i i2 i3 i4 i5 i6 i100 i50 When simplifying imaginary expressions, you need to be careful. Example: Simplify the following a) 16 25 b) 9 81 When multiplying square roots, you can combine and multiply or divide the numbers under the square root together unless both numbers are ______________________. To be safe, always pull the ____ out before taking the square root. When you take the sum of an imaginary number with a real number, you get a _____________________ number. Definition: A _______________________________ is a number of the form ______________, where a and b are real numbers and ________________; a is called the ___________ part and b is called the _______________________ part. Example: Name the real and imaginary parts of 3 4i All properties that hold true for real numbers (except inequalities) hold true for complex numbers as well. Name: Hour: Examples: Mrs. Gorsline Integrated Math 2 Unit 2 Notes: Chapter 6 Add and Simplify Simplify (3 4i) (7 8i) 2i(8 5i) Multiply and Simplify (8 3i)(4 5i) Any number that is a _________________ is not allowed to be in the ______________________ of a fraction. This includes imaginary numbers. Example: Write 3 4i 2 5i in a bi form. Homework (On a separate sheet) p. 391 #7,11-18 and p. 397 #7-10, 16-18 Name: Hour: Mrs. Gorsline Integrated Math 2 Unit 2 Notes: Chapter 6 Factoring Homework Notes There are many types of factoring problems. These are tips that I use when I factor. Problems: 1. 6a4b2 + 9a3b This problem is a binomial so look at the two terms and see if there is a common factor in the two coefficients. If so, write that factor down. Now look at the variables and see if any of them repeat. If so, take the smallest exponent of that variable out. In this problem we would take out ____________. Now we divide this factor into both factors and write the result in parentheses or we can look at it as what times the common factor to get the original problem. Our answer is _________________. Always check your final answer to be sure it can’t be factored any more. 2. Problem 1 process can be applied to x(a + 3) – 4(a + 3) to get___________________. 3. To factor trinomials that don’t have a leading coefficient, follow the following steps. Ex. x4 – 7x3 + 10x2 As in problem 1, factor out any like terms. x2 (x2 –7x + 10) Put two sets of parentheses. x2( )( ) Put the variable inside each parentheses as the first term, (either first or last) x2 (x )(x ). Find numbers that multiply to give you the last coefficient that also add up to get the middle term. In this case -5 times -2 is 10 and -5 + -2 is -7, so place the factors in the appropriate place. x2 (x – 2)(x – 5). Check your work by using FOIL and distributing. Factor the following for Preview Homework: 1. 19x3 - 19x 7. x2 - 4x + 3 2. 36x3 - 24x2 + 8x 8. x2 - 5x - 24 3. -16x4 - 32x3 - 80x2 9. x2 + x - 90 4. x2 + 2x - 35 10. 6x(x - 4) + 5(x - 4) 5. x2 + 8x - 20 11. (x - 10)11 + x(x - 10) 6. x2 - 6x + 8 12. (9x + 10)-10 + 7x(9x + 10) Name: Hour: Mrs. Gorsline Integrated Math 2 Unit 2 Notes: Chapter 6 Factoring In-Class Notes To factor a trinomial with the leading coefficient is not one, there are multiple ways to do it. I am going to teach you 3 different ways. You do it the way you like best. We will use 2x2 – x – 6 for our examples. Method 1: Factor Sum Method Multiply the coefficients of the first and last term. 2 times -6 = ______. Find the factors of -12 that will add up to give us the coefficient of the second term -1. o Rewrite the original problem but replacing the middle term with the two factors you just found o Now you group the first two terms and the last two terms, and factor out the greatest common factor of each group. o After factoring you should come up with the two parentheses to be the same, which then can be factored out of both terms. o Method 2: The reverse box method Begin just like the last problem where you multiply the coefficients of the first and last term, and then find the factors that will add up to the coefficient of the second term (we already know they are 3 and -4) Now draw your 2x2 box and write in the 4 factors that you have Now you need to factor out the greatest common factors across both rows and down both columns. You have magically found the factors! Method 3: Modified Factor Sum Method Again, begin the same way by multiplying the coefficients of the first and last term, and then find the factors that will add up to the coefficient of the second term (again, we know they are 3 and -4) Take those two factors, and divide them both by the leading coefficient (keep as fractions if they don’t divide evenly) o Now, draw your two empty parentheses, placing x as the first term in both sets, and the numbers you just found as the second term. o If you have any fractions, take the denominator of that fraction and bring it up as the coefficient of the x in that group. You have found your factors! o Name: Hour: Mrs. Gorsline Integrated Math 2 Unit 2 Notes: Chapter 6 Special cases: If you have a binomial where each term is a ________________________ after like terms are taken out. o If there is a plus sign between the two terms, such as x2 + 4, this is not factorable. o If there is a minus sign between the two terms, such as x2 – 4, then put two parentheses with opposite signs and fill in with the square roots of each term. o These are called _____________________. Homework: Complete using whichever factoring method you are most comfortable with. 1. x3 + 3x 9. 6x3 - 17x + 5x 2. 14x5 - 24x4 10. 3x2 + x - 10 3. x2 - 100 11. 8x2 - 2x - 3 4. x2 – 16 12. 16x4 - 49 5. 13. 4x2 + 12x + 5 (x - 3)10 + 9x(x - 3) 6. 9x2 - 12x + 4 14. 6x2 - 7x + 1 7. 25x2 - 9 15. 6x2 - 5x + 1 8. x2 + 2x – 35 16. 3x2 - 7x - 6 Name: Hour: Mrs. Gorsline Integrated Math 2 Unit 2 Notes (Chapter 6) 6-10 Homework Notes 6-10: Analyzing Solutions to Quadratic Equations (p. 402) How many real solutions does a Quadratic Equation Have? Quadratic equation ax bx c 0 are x= 2 Because a and b are real numbers, the numbers _________ and _______ are real so only ______________________ could possibly be nonreal. Therefore: If b2 4ac is positive, then there will be _______________________________________________ If b2 4ac is zero, then there will be _______________________________________________ If b2 4ac is negative, then there will be _______________________________________________ Complete p. 405 #3-5,7 Name: Hour: Mrs. Gorsline Integrated Math 2 Unit 2 Notes (Chapter 6) 6-10 In-Class Notes From last night’s notes, the part of the quadratic formula that determines whether the solution is going to be _________ or ___________________ is the part under the ________________________________, which is __________________. Because this is so important, it is given a special name, the _________________________. ______________________ Theorem Suppose a, b, and c are real numbers with a 0. Then the equation ax 2 bx c 0 has (i) ______________________________________ if b2 4ac 0 (ii) ______________________________________ if b2 4ac 0 (iii) _________________________________________________________ if b2 4ac 0 (remember that _________________________________ mean that they have an ________________ part) Solutions to quadratic (and other) equations that equal ________ are often called _____________. The number ____ allows square roots of _________________ numbers to be considered as __________________ solutions. Example: Determine the nature of the roots of the following equations. Then solve. a. 4 x2 12 x 9 0 b. 2 x 2 3x 4 0 Example: Does Pop Fligh’s Ball ever reach a height of 40 feet? Complete p. 405 #9,10,11,14,15 on a separate sheet. c. 2 x 2 3x 9 0