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Name:_______________________________ Period:____ Chapter 4 Notes Packet on Quadratic Functions and Factoring Notes #15: Graphing quadratic equations in standard form, vertex form, and intercept form. A. Intro to Graphs of Quadratic Equations: y ax 2 bx c 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. Ex: y 5 x 2 y 2 x 2 7 y x 2 x 3 The graph of a quadratic function is a U-shaped curve called a ________________. The maximum or minimum point is called the _____________ Identify the vertex of each graph; identify whether it is a minimum or a maximum. 1.) 2.) Vertex: ( , ) _________ Vertex: ( 3.) ) _________ , ) _________ 4.) Vertex: ( , ) _________ Vertex: ( B. Key Features of a Parabola: , y ax 2 bx c Direction of Opening: When a 0 , the parabola opens ________: When a 0 , the parabola opens ________: Width: When a 1 , the parabola is _______________ than y x 2 When a 1 , the parabola is the ________ width as y x 2 When a 1 , the parabola is ________ than y x 2 Vertex: The highest or lowest point of the parabola is called the vertex, which is on the axis of symmetry. To find the vertex, plug in x b and solve for y. This yields a point (____, ____) 2a Axis of symmetry: This is a vertical line passing through the vertex. Its equation is: x b 2a x-intercepts: are the 0, 1, or 2 points where the parabola crosses the x-axis. Plug in y = 0 and solve for x. y-intercept: is the point where the parabola crosses the y-axis. Plug in x = 0 and solve for y. 1 Without graphing the quadratic functions, complete the requested information: 5 5.) f ( x) 3x 2 7 x 1 6.) g ( x) x 2 x 3 4 What is the direction of opening? _______ What is the direction of opening? _______ Is the vertex a max or min? _______ Is the vertex a max or min? _______ 2 Wider or narrower than y = x2 ? ___________ Wider or narrower than y = x ? __________ 7.) y 8.) y 0.6 x 2 4.3x 9.1 2 2 x 11 3 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 ? ___________ The parabola y = x2 is graphed to the right. Note its vertex (___, ___) and its width. You will be asked to compare other parabolas to this graph. C. Graphing in STANDARD FORM ( y ax 2 bx c ): we need to find the vertex first. Vertex Graphing - list a = ____, b = ____, c = ____ - put the vertex you found in the center of your x-y chart. b - find x = - choose 2 x-values less than and 2 x-values more 2a than your vertex. - plug this x-value into the function (table) - this point (___, ___) is the vertex of the parabola - 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 9.) f(x)= -x2 + 2x + 3 a = ____, b = ____, c = ____ y 10 9 8 7 6 5 4 3 Vertex: _______ Max or min? _______ Direction of opening? _______ Axis of symmetry: ________ Compare to the graph of y = x2 _________________________ 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 2 10.) h(x) = 2x2 + 4x + 1 y 10 9 8 7 Vertex: _______ Max or min? _______ Direction of opening? _______ Axis of symmetry: ________ Compare to the graph of y = x2 _________________________ 6 5 4 3 2 1 x 1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1 2 3 4 5 6 7 8 9 10 -2 -3 -4 -5 -6 -7 -8 -9 -10 11.) k(x) = 2 – x – 1 2 x 2 y 10 9 8 7 6 Vertex: _______ Max or min? _______ Direction of opening? _______ Axis of symmetry: ________ Compare to the graph of y = x2 _________________________ 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 12.) State whether the function y = 3x2 + 12x 6 has a minimum value or a maximum value. Then find the minimum or maximum value. 1 2 x 5 x 7 . State whether it is a minimum or maximum. Find that 2 minimum or maximum value. 13.) Find the vertex of y 3 Another useful form of the quadratic function is the vertex form: ________________________________. GRAPH OF VERTEX FORM y = a(x h)2 + k The graph of y = a(x h)2 + k is the parabola y = ax2 translated ___________ h units and ___________ k units. The vertex is (___, ___). The axis of symmetry is x = ___. The graph opens up if a ___ 0 and down if a ___ 0. Find the vertex of each parabola and graph. 2 13.) y x 1 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 Vertex: _______ -7 -8 -9 -10 1 2 14.) y x 1 5 3 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 Vertex: _______ -7 -8 -9 -10 15.) Write a quadratic function in vertex form for the function whose graph has its vertex at (-5, 4) and passes through the point (7, 1). 4 GRAPH OF INTERCEPT FORM y = a(x p)(x q): Characteristics of the graph y = a(x p)(x q): The x-intercepts are ___ and ___. The axis of symmetry is halfway between (___, 0) and ( ___ , 0) and it has equation x = 2 The graph opens up if a ___ 0 and opens down if a ___ 0. 16.) Graph y = 2(x 1)(x 5) 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 x-intercepts: _______, _______ -6 -7 -8 -9 Vertex: _______ -10 Converting between forms: From intercept form to standard form Use FOIL to multiply the binomials together Distribute the coefficient to all 3 terms From vertex form to standard form Re-write the squared term as the product of two binomials Use FOIL to multiply the binomials together Distribute the coefficient to all 3 terms Add constant at the end Ex: y 2 x 5 x 8 Ex: f x 4 x 1 9 2 5 Notes 16: Sections 4-3 and 4-4: Solving quadratics by Factoring A. Factoring Quadratics Examples of monomials:_______________________________ Examples of binomials:________________________________ Examples of trinomials:________________________________ Strategies to use: (1) Look for a GCF to factor out of all terms (2) Look for special factoring patterns as listed below (3) Use factor by grouping (4) Check your factoring by using multiplication/FOIL Factor each expression completely. Check using multiplication. 1.) 3 x 2 15 x 2.) 6 x 2 24 3.) x 2 5 x 24 4.) 25 x 2 81 6.) 4 x 2 12 x 9 5.) m2 22m 121 6 7.) 5 x 2 17 x 6 8.) 3x 2 5 x 12 9.) 25t2 110t + 121 10.) 16 x 2 36 11.) 9a 2 42a 49 12.) 6 x 2 33x 36 B. Solving quadratics using factoring To solve a quadratic equation is to find the x values for which the function is equal to _____. The solutions are called the _____ or _______of the equation. To do this, we use the Zero Product Property: Zero Product Property List some pairs of numbers that multiply to zero: (___)(___) = 0 (___)(___) = 0 (___)(___) = 0 (___)(___) = 0 What did you notice? _______________________________________________ ZERO PRODUCT PROPERTY If the _________ of two expressions is zero, then _______ or _______ of the expressions equals zero. Algebra If A and B are expressions and AB = ____ , then A = _____ or B = __. Example If (x + 5)(x + 2) = 0, then x + 5 = 0 or x + 2 = 0. That is, x = __________ or x = _________. Use this pattern to solve for the variable: 1. get the quadratic = 0 and factor completely 2. set each ( ) = 0 (this means to write two new equations) 3. solve for the variable (you sometimes get more than 1 solution) 7 Find the roots of each equation: 13.) x 2 7 x 30 0 15.) 3x 2 2 x 21 6 4 8 2 14.) x x 0 7 5 9 3 Find the zeros of each equation: 17.) v(v + 3) = 10 16.) 2 x 2 8 x 30 x 34 18.) 2 x 2 x 15 Find the zeros of the function by rewriting the function in intercept form: 19.) y 6 x 2 3x 63 20.) f x 12x2 6x 6 21.) g x 49x2 16 Graph the function. Label the vertex and axis of symmetry: 22.) y 2 x 1 ( x 3) y 10 9 8 7 6 5 4 Vertex: _______ Maximum or minimum value: _______ x-intercepts: _______ Axis of symmetry: ________ Compare width to the graph of y = x2 _________________________ 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 8 D. 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) 8.) x2 = 25 10.) 4x2 – 1 = 0 12.) (2y + 3)2 = 49 9.) 3x2 = 81 11.) m2 3 2 15 13.) 3 x 2 6 34 2 9 Section 4.7: Completing the Square B. Review: Solving Using Square Roots Factor and write one side of the equation as the square of a binomial Square root both sides of the equation (include + and – on the right side; 2 equations! Solve for the variable (make sure there are no roots in the denominator) 1) (k + 2)2 = 12 2.) x2 + 2x + 1 = 8 3.) n2 – 14n + 49 = 3 C. Completing the Square ax 2 bx c 0 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 4.) x2 + 6x + _____ ( )( ( 5.) m2 – 14m + _______ ) )2 Find the value of c such that each expression is a perfect square trinomial. Then write the expression as the square of a binomial. 6.) w2 + 7w + c 7.) k2 – 5k + c 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 8.) x2 + 4x – 5 = 0 9.) m2 – 5m + 11 = 10 10 10.) 2k 2 9k 3 k 17 11.) 3w2 4w 11 2 2 w 12.) 2x2 – 3x – 1 = 0 13.) 3x 2 x 2 x 6 Cumulative Review: Solving Quadratics Solve by factoring: 14.) 12k2 – 5k = 2 15.) 49m2 – 16 = 0 Solve by using square roots: 16.) 4w2 = 18 17.) 3y2 – 8 = 0 11 Notes #19: Sec. 4-8 Use the Quadratic Formula and the Discriminant A. Review of Simplifying Radicals and Fractions Simplify expression under the radical sign; reduce Reduce only from ALL terms of the fraction. (You can’t reduce a number outside of a radical with a number inside of a radical) Make sure that you have TWO answers Simplify: 6 18 1.) 2 2.) 5 20 2 3.) 4 20 4 4.) 8 27 2 5.) 9 (5)2 (5)(2)(3) 4 6.) 9 (6)2 4(3)(3) 4 B. Solving Quadratics using the Quadratic Formula So far, we have solved quadratics by: (1) _______________, (2) ______________, and (3) ___________________. 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 (look for 12 1 i ); reduce Solve by using the quadratic formula: 2 1.) x + x = 12 x b b2 4ac 2a (std. form): a = _____ b = _____ c = _____ 2.) 5x – 8x = -3 2 b b2 4ac x 2a (std. form): a = _____ b = _____ c = _____ 3.) -x2 + x = -1 4.) 3x2 = 7 – 2x 5.) -x2 + 4x = 5 6.) 4( x 1)2 6 x 2 13 C. Using the Discriminant Quadratic equations can have two, one, or no solutions (x-intercepts). You can determine how many solutions a quadratic equation has before you solve it by using the ________________. b b2 4ac The discriminant is the expression under the radical in the quadratic formula: x 2a Discriminant = b2 – 4ac If b2 – 4ac < 0, then the equation has 2 imaginary solutions If b2 – 4ac = 0, then the equation has 1 real solution If b2 – 4ac > 0, then the equation has 2 real 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 and type of solutions Find the discriminant of the quadratic equation and give the number and type of solutions of the equation. 3.) 3 x 2 5 x 1 4.) x 2 3 x 7 5.) 9x2 – 6x = 1 6.) 4x2 = 5x + 3 14 Cumulative Review Problems: Solve by factoring: 7.) 4m2 +5m – 6 = 0 8.) 3x3 – 27x = 0 Solve by using square roots: 9.) 4b2 + 1 = 0 10.) (3x + 1)2 = 18 Solve by completing the square: 11.) 4m2 + 12m +5 = 0 12.) x2 – 7x – 18 = 0 For #13, find the vertex of the parabola. Graph the function and find the requested information 13.) g(x) = -2x2 + 8x – 5 y 10 9 8 7 6 5 Vertex: _______ 4 3 2 Max or min value: _______ 1 x -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1 1 2 3 4 5 6 7 8 9 10 Direction of opening? _______ -2 -3 Compare width to y = x2 : -4 -5 -6 _______________ -7 -8 -9 Axis of symmetry: ________ -10 15 Notes #20: Section 4-9 Graph and Solve Quadratic Inequalities GRAPHING A QUADRATIC INEQUALITY IN TWO VARIABLES To graph a quadratic inequality, follow these steps: Step 1 Graph the parabola with equation y = ax2+ bx + c. Make the parabola _______________ for inequalities with < or > and ______________ for inequalities with or . Step 2 Test a point (x, y) ______________ the parabola to determine whether the point is a solution of the inequality. Step 3 Shade the region ______________ the parabola if the point from Step 2 is a solution. Shade the region ________________ the parabola if it is not a solution. 1.) y x2 2x + 3 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 2.) y ≥ 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 16 1 2 3 4 5 6 7 8 9 10 3.) y > 3x2 +3x 5 y x2 +5x + 10 4.) y -x2 +4 y x2 − 2x− 3 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 17 1 2 3 4 5 6 7 8 9 10 Notes# 21: Sec. 4-10 Write Quadratic Functions and Models A. When given the vertex and a point Plug the vertex in for (h, k) in y a( x h)2 k Plug in the given point for (x, y) Solve for a. Plug in a, h, k into y a( x h)2 k 1.) Write a quadratic equation in vertex form for the parabola shown. 2.) Write a quadratic function in vertex form for the function whose graph has its vertex at (2, 1) and passes through the point (4, 5). B. When given the x-intercepts and a third point Plug in the x-intercepts as p and q into y = a(x p)(x q) Plug in the given point for (x, y) Solve for a. Plug in a, h, k into y = a(x p)(x q) 3.) Write a quadratic function in intercept form for the parabola shown. 18 Notes #22: Chapter 4 Review To graph a quadratic function, you must FIRST find the vertex (h, k)!! (A) If the function starts in standard form y ax 2 bx c : b 1st: The x-coordinate of the vertex, h, = 2a 2nd: Find the y-coordinate of the vertex, k, by plugging the x-coordinate into the function & solving for y. (B) If the function starts in intercept form y a( x p)( x q ) : 1st: Find the x-intercepts by setting the factors with x equal to 0 & solving for x. 2nd: The x-coordinate of the vertex is half way between the x-intercepts. 3rd: Find the y-coordinate of the vertex, k, by plugging the x-coordinate into the function & solving for y. (C) If the function starts in vertex form y a( x h)2 k : 1st: pick out the x-coordinate of the vertex, h. REMEMBER: h will have the OPPOSITE sign as what is in the parenthesis!! nd 2 : Pick out the y-coordinate of the vertex, k. It will have the SAME sign as the what is in the equation! Direction of Opening: If a is positive, the graph opens up If a is negative, the graph opens down. Width of the function: If a 1 , the graph is narrower than y x 2 If a 1 , the graph is wider than y x 2 19 Graph each function. (A) State the vertex. (B) State the direction of opening (up/down). (C) State whether the graph is wider, narrower, or the same width as y x 2 . 1 1.) f ( x) ( x 6) 2 5 2 y 10 9 8 7 6 Vertex: _______ 5 4 3 Direction of opening? _______ 2 1 x -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1 1 2 3 4 5 6 7 8 9 10 Compare width to the graph of y = x2 _________________________ -2 -3 -4 -5 -6 -7 -8 -9 -10 2.) k(x) = x2 + 2x + 1 y 10 9 8 7 6 5 4 Vertex: _______ 3 2 1 Direction of opening? _______ x -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1 1 2 3 4 5 6 7 8 9 10 -2 -3 Compare width to the graph of y = x2 _________________________ -4 -5 -6 -7 -8 -9 -10 3.) f(x) = x2 – x – 6 y 10 9 8 7 6 Vertex: _______ 5 4 3 Direction of opening? _______ 2 1 x -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1 1 2 3 4 5 6 7 8 9 10 Compare width to the graph of y = x2 _________________________ -2 -3 -4 -5 -6 -7 -8 -9 -10 20 4.) f ( x) x2 2 x 3 y 10 9 8 7 Vertex: _______ 6 5 4 3 Direction of opening? _______ 2 1 x -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1 1 2 3 4 5 6 7 8 Compare width to the graph of y = x2 _________________________ 9 10 -2 -3 -4 -5 -6 -7 -8 -9 -10 5.) h( x) 2 x 2 4 x 1 y 10 9 8 7 Vertex: _______ 6 5 4 3 Direction of opening? _______ 2 1 x -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1 1 2 3 4 5 6 7 8 9 10 Compare width to the graph of y = x2 _________________________ -2 -3 -4 -5 -6 -7 -8 -9 -10 1 6.) g ( x) ( x 4)( x 6) 3 y 10 9 8 Vertex: _______ 7 6 5 4 Direction of opening? _______ 3 2 1 Compare width to the graph of y = x2 _________________________ 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 21 Methods for Solving Quadratic Equations: A.) Factoring 1st: Set equal to 0 2nd: Factor out the GCF 3rd: Complete the X & box method to find the factors 4th: Set every factor that contains an x in it, equal to 0 & solve for x. B.) Completing the Square 1st: Move the constant (number with no variable) to the right so that all variables are on the left & all constants are on the right. nd 2 : Divide every term in the equation by the value of a, if it is not already 1. 2 b 3rd: Create a perfect square trinomial on the left side by adding to both sides. 2 2 b 4 : Factor the left side into a x and simplify the value on the right side. 2 th 5 : Take the square root of both sides of the equation. REMINDER: Don’t forget the 6th: Solve for x th C.) Finding Square Roots 1st: Isolate the term with the square. 2nd: Take the square root of both sides of the equation. REMINDER: Don’t forget the 3rd: Solve for x. D.) Quadratic Formula 1st: Set the equation equal to 0. 2nd: Find the values of a, b, and c & plug them into the Quadratic Formula: b b2 4ac 2a rd 3 : Simplify the radical as much as possible. 4th: If possible, simplify the numerator into integers. x 46 3 ), divide BOTH 2 terms by the number in the denominator (the example would result in 2 3 3 ) 5th: Divide. REMINDER: If you have 2 terms in the numerator (ex: Examples: Solve each equation by the method stated. By Square Roots: 1.) (2y + 3) 2 = 49 2.) (3m – 1) 2 = 20 22 3.) (3r 5)2 1 11 By Factoring: 4.) x2 – 4x – 5 = 0 5.) 3 x 2 2 x 21 6.) 2 x 2 x 15 By Completing the Square: 7.) x2 – 6x – 11 = 0 8.) 2y2 + 6y – 18 = 0 9.) 2x2 – 3x – 1 = 0 By Quadratic Formula: 10.) x2 + x = 12 11.) 5x2 – 8x = -3 12.) 2x2 = 4 – 7x 23 Chapter 4 Review Sheet Please complete each problem on a separate sheet of paper. Show all of your work and please use graph paper for all graphs. For questions 1-3, solve by factoring. 1. x2 12 x 32 0 2. 3x2 8x 5 0 3. 2x2 4x 30 0 For questions 4-5, solve by finding square roots. 1 2 4. ( x 1) 2 16 0 5. 2 x 1 3 6 4 For questions 6-8, solve by completing the square. 6. x2 16x 15 0 1 8. x 2 4 x 6 0 2 7. 5x2 10 x 20 0 For questions 9-10, solve by using the quadratic formula. 9. x2 3x 5 0 10. 3x2 2 x x2 5x 1 For questions 11-13, simplify each expression. 48 32 5 11. 12. 4 48 13. 252 3 For questions 14-16, write each function in vertex form, graph the function, and label the vertex and axis of symmetry. 1 14. y x 2 16 x 2 15. y 2 x 2 4 x 7 16. y ( x 4) 2 8 3 For question 17-19, graph each inequality. 17. y 2( x 1)2 5 18. y x 2 8 x 2 19. y 2 x 2 12 x 16 For questions 20-21, write a quadratic function in vertex form whose graph has the given vertex and passes through the given point. 20. Vertex (2, 3) and passes through (-3, 7) 21. Vertex (-1, -5) and passes through (2, -1) For questions 22-23, write a quadratic function in standard form whose graph passes through the given points. 22. (1, 1), (0, -2), (2, 8) 23. (-1, -7), (1, -5), (2, -1) For questions 24-28, write the expression as a complex number in standard form. 24. (7 3i) (9 4i) 25. (6 2i)(8 3i) 26. (3 7i ) (9 2i) 27. 2 5i 7i 28. 4 7i 3i Chapter 4 Test and Notes Check Tomorrow! 24