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Name: Date:__________ Period: __________ CHAPTER 9: Quadratic Equations and Functions Notes #23 9-1: Exploring Quadratic Graphs A. Graphing y ax 2 A ____________________ is a function that can be written in the form y ax 2 bx c where a, b, and c are real numbers and a 0. Examples: y 5 x 2 y x2 x 3 The graph of a quadratic function is a U-shaped curve called a ________________. When graphed it will look like: y x2 7 OR You can fold a parabola so that the two sides match exactly. This property is called: _____________. The highest or lowest point of the parabola is called the ________________, which is on the axis of symmetry. B. Identifying a Vertex Identify the vertex of each graph. Tell whether it is a minimum or a maximum. 1.) Vertex: ( 2.) , ) _________ 3.) Vertex: ( Vertex: ( , ) _________ , ) _________ 4.) , ) _________ Vertex: ( Algebra 1 Notes: Chapter 9 -2Graph each function. State the domain, the vertex (min/max point), the range, the x-intercepts, and the axis of symmetry. 5.) f(x)= x2 – 4 y 10 9 8 7 6 5 4 3 2 1 Domain: ________ Range: ________ Vertex:________ Max or min?________ x-intercepts: ____________ Axis of symmetry: ________ x -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1 1 2 3 4 5 6 7 8 9 10 -2 -3 -4 -5 -6 -7 -8 -9 -10 6.) h(x) = -2x2 y 10 9 8 7 6 5 4 3 2 1 x -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1 1 2 3 4 5 6 7 8 9 10 Domain: ________ Range: ________ Vertex:________ Max or min?________ x-intercepts: ____________ Axis of symmetry: ________ -2 -3 -4 -5 -6 -7 -8 -9 -10 7.) f(x) = 1 2 x 2 y 10 9 8 7 6 5 4 3 2 1 x -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 1 2 3 4 5 6 7 8 9 10 Domain: ________ Range: ________ Vertex:________ Max or min?________ x-intercepts: ____________ Axis of symmetry: ________ Algebra 1 Notes: Chapter 9 -38.) k(x) = x2 + 2x + 1 y 10 9 8 7 6 5 4 3 2 1 x -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1 1 2 3 4 5 6 7 8 9 10 3 4 5 6 7 8 9 10 Domain: ________ Range: ________ Vertex:________ Max or min?________ x-intercepts: ____________ Axis of symmetry: ________ -2 -3 -4 -5 -6 -7 -8 -9 -10 9.) f(x) = x2 – x - 6 y 10 9 8 7 6 5 4 3 2 Domain: ________ Range: ________ Vertex:________ Max or min?________ x-intercepts: ____________ Axis of symmetry: ________ 1 x -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1 1 2 -2 -3 -4 -5 -6 -7 -8 -9 -10 C. Comparing Widths of parabolas The value of a, the coefficient of the x 2 term in a quadratic function, affects the width of the parabola as well as the direction in which it opens. When a 1, then the parabola is steeper, (or _________) than y = x2 When a 1, then the parabola is not as steep, (or _________) than y = x2 Order each group of quadratic functions from widest to narrowest graph: 10.) f ( x) 3x 2 , f ( x) 4 x 2 , f ( x) 1 2 x 2 5 11.) y 2 x 2 , y x 2 , y x 2 4 Algebra 1 Notes: Chapter 9 -4D. Applications 12.) A monkey drops an banana from a branch 64 feet above the ground. Gravity causes the banana to fall. The function h 16t 2 64 gives the height of the banana, h, in feet, after t seconds. a) Graph this quadratic function b) When does the banana hit the ground? t h(t ) 16t 2 64 (t, h(t)) 70 60 50 feet 40 30 20 10 1 2 3 time (sec) 13) A bungee jumper dives from a platform. The function h = -16t2 + 160 describes her height, h, after t seconds in the air. a) What will her height be after 1 second? b) what will her height be after 2 seconds? c) How far did she fall between 1 and 2 seconds in the air? Algebra 1 Notes: Chapter 9 -5Notes #24 9-2: Quadratic Functions y = ax2 + bx + c One key characteristic of a parabola is its vertex (min/max point). Yesterday we found the vertex after we graphed the function. It would help to find the vertex first. Vertex - find x = b 2a - plug this x-value into the function (table) - this point (___, ___) is the vertex of the parabola Graphing - put the vertex you found in the center of your x-y chart. - choose 2 x-values less than and 2 x-values more than your vertex. - plug in these x values to get 4 more points. - graph all 5 points Find the vertex of each parabola. Graph the function and find the requested information 1.) f(x)= -x2 + 2x + 3 a = ____, b = ____, c = ____ y 10 9 8 7 6 5 4 3 2 1 x -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1 1 2 3 4 5 6 7 8 9 10 -2 -3 -4 -5 -6 -7 -8 -9 -10 Vertex: _______ Max or min? _______ Direction of opening? _______ Wider or narrower than y = x2 ? _______________ Domain: ________ Range: ________ x-intercepts: ____________ Axis of symmetry: ________ 2.) h(x) = 2x2 + 4x + 1 y 10 9 8 7 6 5 4 3 2 1 x -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 1 2 3 4 5 6 7 8 9 10 Vertex: _______ Max or min? _______ Direction of opening? _______ Wider or narrower than y = x2 ? _______________ Domain: ________ Range: ________ x-intercepts: ____________ Axis of symmetry: ________ Algebra 1 Notes: Chapter 9 -63.) k(x) = 2 – x – 1 2 x 2 y 10 9 8 7 6 5 4 3 2 1 x -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1 1 2 3 4 5 6 7 8 9 10 -2 -3 -4 -5 -6 -7 -8 -9 -10 Vertex: _______ Max or min? _______ Direction of opening? _______ Wider or narrower than y = x2 ? _______________ Domain: ________ Range: ________ x-intercepts: ____________ Axis of symmetry: ________ Without graphing the quadratic functions, complete the requested information: 4.) f ( x) 3x 2 7 x 1 5 5.) g ( x) x 2 x 3 4 What is the direction of opening? _______ Is the vertex a max or min? _______ Wider or narrower than y = x2 ? ___________ What is the direction of opening? _______ Is the vertex a max or min? _______ Wider or narrower than y = x2 ? ___________ 6.) y 2 2 x 11 3 What is the direction of opening? _______ Is the vertex a max or min? _______ Wider or narrower than y = x2 ? ___________ 7.) y 0.6 x 2 4.3x 9.1 What is the direction of opening? _______ Is the vertex a max or min? _______ Wider or narrower than y = x2 ? ___________ B. Application 8.) Suppose a particular star is projected from an aerial firework at a starting height of 520 feet with an initial upward velocity of 88 ft/s. How long will it take for the star to reach its maximum height? How far above the ground will it be? The equation h 16t 2 88t 520 gives the star’s height h in feet at time t in seconds. Algebra 1 Notes: Chapter 9 -7Notes #25 9-3: Finding and Estimating Square Roots A. Finding Square Roots The expression __________ means the positive, or __________ square root. The expression __________ means the negative square root. The expression ___________ means both the ____________ and _____________ square root Simplify each expression. 1.) 4.) 0 5.) 7) 8.) 72 27 10.) 3.) 2.) 100 64 4 5 6.) 0.09 25 11.) 9 16 9.) 108 45 4 12.) 2.25 B. Rational and Irrational Square Roots Tell whether each expression is rational or irrational. 13.) 144 14.) 1 5 17.) Between what two consecutive integers is 15.) 28.34 ? 1 9 16.) 7 Algebra 1 Notes: Chapter 9 -818.) Between what two consecutive integers is 68.7 ? 19.) Between what two consecutive integers is 14.3 ? C. Application: Pythagorean Theorem (Review) Use the Pythagorean theorem (_________________) to solve for the missing side of the right triangle. 20.) 21.) 6 4 8 x y 4 9-4: Solving Quadratic Equations A. Solving Quadratic Equations by Graphing The solutions of a quadratic equation and the related x-intercepts are often called _______ of the equation or _______ of the function. 1.) The function f(x) = x2 + x – 6 is graphed to the left. a) Circle and name the zeros of the function graphed here. ( , ) and ( , ) b) Use this graph to solve the equation: x2 + x – 6 = 0 (This is asking: “At what x-values does y = 0?”) Algebra 1 Notes: Chapter 9 -9B. Solving by Graphing Solve each equation by graphing the related function: Find the vertex and 4 other points on the parabola; graph. Find the x-intercepts from the graph. These are the _______ or _______. 2.) x2 – 4 = 0 y 10 9 8 7 6 5 4 3 2 1 x -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 6 7 8 9 10 -2 -3 -4 -5 -6 -7 -8 -9 -10 3.) 2x2 – 2 = 0 y 10 9 8 7 6 5 4 3 2 1 x -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 4.) x2 + 6 = 0 y 10 9 8 7 6 5 4 3 2 1 x -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 1 2 3 4 5 Algebra 1 Notes: Chapter 9 - 10 C. Solving Quadratic Equations Using Square Roots Isolate the variable or expression being squared (get it ______________) Square root both sides of the equation (include + and – on the right side!) This means you have _____________ equations to solve!! Solve for the variable (make sure there are no roots in the denominator) 5.) x2 = 25 6.) 3x2 = 48 7.) 4x2 – 1 = 0 8.) 3m2 – 5 = 0 9.) 2y2 – 81 = 0 10.) 36b2 – 7 = 0 11.) (x – 1)2 = 4 12.) (2y + 3)2 = 49 13.) (r + 5)2 = 12 14.) (3m – 1)2 = 20 Algebra 1 Notes: Chapter 9 - 11 If the left side is not already factored or squared, _______________ it! 15.) x2 + 2x + 1 = 8 16.) n2 – 14n + 49 = 3 17.) w2 + 22w + 121 = 169 18.) g2 + 10g + 25 = 18 D. Application 19.) A museum is planning an exhibit that will contain a large globe. The surface area of the globe will be 100 ft2. Find the radius of the sphere producing this surface area. Use the equation S 4 r 2 , where S is the surface area and r is the radius. Algebra 1 Notes: Chapter 9 - 12 Notes #26 9-5: Solving Quadratic Equations by Factoring A. Solving Quadratic Equations Zero Product Property List some pairs of numbers that multiply to zero: (___)(___) = 0 (___)(___) = 0 (___)(___) = 0 (___)(___) = 0 What did you notice? _______________________________________________ Use this pattern to solve for the variable: 1. get = 0 and factor (sometimes this is done for you) 2. set each ( ) = 0 (this means to write two new equations) 3. solve for the variable (you sometimes get more than 1 solution) 1.) (3)(x) = 0 2.) (2)(x + 1) = 0 4.) (m + 1)(5m – 3) = 0 5.) 7.) x2 – 4x – 5 = 0 8.) y2 + 6 + 5y = 0 3 w 2w 9 0 5 3.) -2y(y – 7) = 0 6 4 8 2 6.) x x 0 7 5 9 3 9.) 4v2 – 9 = 0 10.) x 2 x 42 0 13.) v(v + 3) = 10 11.) 3x 2 2 x 21 Algebra 1 Notes: Chapter 9 - 13 2 12.) 2 x x 15 14.) b(b – 2) = 3(b + 2) B. Solving Word Problems with Quadratics Steps: 1. Draw a picture and define your variable (let statement) 2. Write an equation 3. Get = 0 (bring all variables and numbers to one side) 4. Factor completely and solve 5. Do all the answers make sense? 6. Write your answer in a complete sentence Translate and solve: 15.) The square of a positive number minus twice the number is 48. Find the number. Let n = _____________ _________ - _________ = ______ 16.) One more than a negative number times one less than that number is 8. Find the number. Let n = ______________ (_________)(_________) = _____ Algebra 1 Notes: Chapter 9 - 14 17.) The product of two consecutive integers is 12. Find the integers. Let x = 1st integer ______ = 2nd integer 18.) The product of two consecutive odd integers is 35. Find the integers. Let x = 1st odd integer ______ = 2nd odd integer 19.) The length of a rectangle is 3ft greater than its width. The area of the rectangle is 54ft2. Find the length and the width of the rectangle. 20.) The area of a square is 5 more square inches than there are inches in the square’s perimeter. Find the length of a side of the square. Algebra 1 Notes: Chapter 9 - 15 21.) Two less than the square of a number is equal to the number. Find the number. ___________ - _________ = _______ 22.) The sum of the square of a number and three times the number is the same as one less than the number. Find the number. __________ + _________ = _________ - _________ Notes #27 9-6: Completing the Square So far in this course, we have solved quadratics by _______________, __________________ and ___________________. We will eventually learn two more ways to solve quadratics. Solve these equations. What makes these quadratics “easy” to solve? a) (x – 1)2 = 9 b) (k + 2)2 = 12 Solving quadratics by _________________ ______ _______________ helps us turn all quadratics into this form. Complete the square: Take half the b (the x coefficient) Square this number (no decimals – leave as a fraction!) Add this number to the expression Factor – it should be a binomial, squared ( )2 1.) x2 + 6x + _____ ( )( ( )2 2.) m2 – 14m + _______ ) Algebra 1 Notes: Chapter 9 - 16 Find the value of n such that each expression is a perfect square trinomial: 3.) w2 + 7w + n 4.) k2 – 5k + n 5.) j2 – j + n 6.) y2 + 18y + n Solving by Completing the Square: Collect variables on the left, numbers on the right Divide ALL terms by a; leave as fractions (no decimals!!) Complete the square on the left – add this number to BOTH sides Square root both sides (include a ______ and _______ equation!) Solve for the variable (simplify all roots) 7.) x2 + 4x – 5 = 0 8.) x2 – 6x – 11 = 0 9.) k2 – 4k – 7 = 0 10.) m2 – 5m + 1 = 0 11.) 2y2 + 6y – 18 = 0 12.) 2x2 – 3x – 1 = 0 Worksheet DA #29: Cumulative Review: Solving Quadratics Algebra 1 Notes: Chapter 9 - 17 Name__________________ Solve by factoring: 1.) 12k2 – 5k = 2 2.) 49m2 – 16 = 0 Solve by using square roots: 3.) 4w2 = 18 4.) 3y2 – 8 = 0 5.) 5m2 – 16 = 0 6.) (2x – 1)2 = 20 Solve by completing the square: 7.) x2 – 10x + 7 = 0 8.) 4m2 + 12m – 7 = 0 9.) 3y2 – 2y – 1 = 0 10.) 2x2 – 20x = -50 11.) 5x2 + 13x + 7 = 0 12.) ax2 + bx + c = 0 Algebra 1 Notes: Chapter 9 - 18 - Notes #28 9-7: Using the Quadratic Formula A. Review of Simplifying Radicals and Fractions Simplify expression under the radical sign, reduce Reduce only from ALL terms of the fraction 1.) 6 18 2 2.) 5 20 2 3.) 4 20 4 4.) 8 27 2 9 (5)2 (4)(2)(3) 5.) 4 9 (6)2 4(3)(3) 6.) 4 Algebra 1 Notes: Chapter 9 - 19 B. Solving Quadratics using the Quadratic Formula So far, we have solved quadratics by: (1) _______________, (2) ______________, (3) ___________________, and (4) _________________ The final method for solving quadratics is to use the quadratic formula. Solving using the quadratic formula: Put into standard form (ax2 + bx + c = 0) List a = , b = , c = b b2 4ac x 2a Plug a, b, and c into Simplify all roots, reduce Solve by using the quadratic formula: 1.) x2 + x = 12 b b2 4ac x 2a (std. form): a = _____ b = _____ c = _____ 2.) 5x2 – 8x = -3 (std. form): a = _____ b = _____ c = _____ b b2 4ac x 2a Algebra 1 Notes: Chapter 9 - 20 3.) 2x2 = 4 – 7x 4.) 3x2 – 8 = 10x 5.) -x2 + x = -1 6.) 3x2 = 7 – 2x Review of Solving Quadratics: Solve by factoring: 7.) 4m2 +5m – 6 = 0 8.) 3x3 – 27x = 0 Solve by using square roots: 9.) 4b2 – 1 = 0 10.) 3y2 – 36 = 0 11.) (3x + 1)2 = 18 Algebra 1 Notes: Chapter 9 - 21 Solve by completing the square: 12.) x2 – 10x + 9 = 0 12.) x2 – 7x – 18 = 0 13.) 4m2 + 12m + 5 = 0 14.) 3y2 + 2y – 1 = 0 Solve by using the quadratic formula: 15.) x2 – 20 = 0 16.) x2 – 6x + 9 = 0 Algebra 1 Notes: Chapter 9 - 22 Notes #29 9-8: Using the Discriminant Quadratic equations can have two, one, or no solutions. You can determine how many solutions a quadratic equation has before you solve it by using the ________________. The discriminant is the expression under the radical in the quadratic formula: x b b2 4ac 2a Discriminant = b2 – 4ac If b2 – 4ac = 0, then the equation has 1 solution If b2 – 4ac < 0, then the equation has 0 real solutions If b2 – 4ac > 0, then the equation has 2 solutions A. Finding the number of x-intercepts Determine whether the graphs intersect the x-axis in zero, one, or two points. 1.) y 4 x 2 12 x 9 2.) y 3x 2 13x 10 B. Finding the number of solutions Find the number of solutions for the following: 3.) 3 x 2 5 x 1 4.) x 2 3 x 7 5.) 9x2 – 6x = 1 6.) 4x2 = 5x + 3 Algebra 1 Notes: Chapter 9 - 23 - C. Review of Solving Quadratics Solve by factoring: 7.) 2x2 + 12x = -10 8.) 16(x – 1) = x(x + 8) Solve by using square roots: 9.) 3b2 – 1 = 7 10.) (3x + 1)2 = 18 Solve by completing the square: 11.) x2 – 10x – 11 = 0 12.) x2 – 3x – 6 = 0 Algebra 1 Notes: Chapter 9 - 24 Solve using the quadratic formula: 13.) 14.) x2 = 8 – 6x 6x2 + 7x = 5 Graph the quadratic. Name the vertex, axis of symmetry, x-intercepts, domain, and range. 15.) f(x)= x2 – 9 y 10 9 8 7 6 5 4 3 2 1 x -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 1 2 3 4 5 6 7 8 9 10 Domain: ________ Range: ________ Vertex:________ Max or min?________ x-intercepts: ____________ Axis of symmetry: ________ Algebra 1 Notes: Chapter 9 - 25 Notes #30: Solving Radical Equations with Quadratics (Section 10.4) Solving Radical Equations: Isolate the _________________ ______________ both sides. If one side is a binomial, be sure to use ___________ to square it. Get all terms to one side to = 0 Solve the quadratic using: factoring, quadratic formula, or completing the square. Check your solution by ___________________ into the original equation. Check for extraneous roots. 1.) 2 x 3 1 2.) 4 x 2 5 3.) x2 x 4.) 35 2x x 5.) 6x 9 x 6.) 11x 28 x Algebra 1 Notes: Chapter 9 - 26 7.) 2x 7 x 2 8.) 3x 2 4 x 9.) Graph the quadratic. Find the requested information: f(x)= -x2 – 3x + 4 y 10 9 8 7 6 5 4 3 2 1 x -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 1 2 3 4 5 6 7 8 9 10 Domain: ________ Range: ________ Vertex:________ Max or min?________ x-intercepts: ____________ Axis of symmetry: ________