Unit 2 Notes Packet on Quadratic Functions and Factoring
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1 Name: Period: Unit Notes Packet on Quadratic Functions and Factoring Notes #: Graphing quadratic equations in standard form, verte form, and intercept form. A. Intro to Graphs of Quadratic Equations: a b c A is a function that can be written in the form a b c where a, b, and c are real numbers and a 0. E: The graph of a quadratic function is a U-shaped curve called a. The maimum or minimum point is called the Identif the verte of each graph; identif whether it is a minimum or a maimum..).) Verte: (, ) Verte: (, ).).) Verte: (, ) Verte: (, ) B. Ke Features of a Parabola: a b c Direction of Opening: When a 0, the parabola opens : When a 0, the parabola opens : Width: When a, the parabola is than When a, the parabola is the width as When a, the parabola is than Verte: The highest or lowest point of the parabola is called the verte, which is on the ais of b smmetr. To find the verte, plug in and solve for. This ields a point (, ) a b Ais of smmetr: This is a vertical line passing through the verte. Its equation is: a -intercepts: are the 0,, or points where the parabola crosses the -ais. Plug in = 0 and solve for. -intercept: is the point where the parabola crosses the -ais. Plug in = 0 and solve for.
2 Without graphing the quadratic functions, complete the requested information:.) f ( ).) g( ) What is the direction of opening? What is the direction of opening? Is the verte a ma or min? Is the verte a ma or min? Wider or narrower than =? Wider or narrower than =?.).) 0... What is the direction of opening? Is the verte a ma or min? Wider or narrower than =? What is the direction of opening? Is the verte a ma or min? Wider or narrower than =? The parabola = is graphed to the right. Note its verte (, ) and its width. You will be asked to compare other parabolas to this graph. C. Graphing in STANDARD FORM ( a b c ): we need to find the verte first. Verte Graphing - list a =, b =, c = - put the verte ou found in the center of b our - chart. - find = - choose -values less than and -values more a - plug this -value into the function (table) than our verte. - this point (, ) is the verte of the parabola - plug in these values to get more points. - graph all points Find the verte of each parabola. Graph the function and find the requested information.) f()= a =, b =, c = Verte: Ma or min? Direction of opening? Ais of smmetr: Compare to the graph of =
3 0.) h() = Verte: Ma or min? Direction of opening? Ais of smmetr: Compare to the graph of =.) k() = Verte: Ma or min? Direction of opening? Ais of smmetr: Compare to the graph of =.) State whether the function = + has a minimum value or a maimum value. Then find the minimum or maimum value..) Find the verte of. State whether it is a minimum or maimum. Find that minimum or maimum value.
4 Another useful form of the quadratic function is the verte form:. GRAPH OF VERTEX FORM = a( h) + k The graph of = a( h) + k is the parabola = a translated h units and k units. The verte is (, ). The ais of smmetr is =. The graph opens up if a 0 and down if a 0. Find the verte of each parabola and graph..) ) Verte: Verte:.) Write a quadratic function in verte form for the function whose graph has its verte at (-, ) and passes through the point (, ).
5 GRAPH OF INTERCEPT FORM = a( p)( q): Characteristics of the graph = a( p)( q): The -intercepts are and. The ais of smmetr is halfwa between (, 0) and (, 0) and it has equation = The graph opens up if a 0 and opens down if a 0..) Graph = ( )( ) Converting between forms: From intercept form to standard form Use FOIL to multipl the binomials together Distribute the coefficient to all terms -intercepts:, Verte: E: From verte form to standard form Re-write the squared term as the product of two binomials Use FOIL to multipl the binomials together Distribute the coefficient to all terms Add constant at the end E: f HW #: Pg. 0: - odd pg. 0 #- s pg. #-0 s
6 Notes : Sections - and -: Solving quadratics b Factoring A. Factoring Quadratics Eamples of monomials: Eamples of binomials: Eamples of trinomials: Strategies to use: () Look for a GCF to factor out of all terms () Look for special factoring patterns as listed below () Use the X-Bo method () Check our factoring b using multiplication/foil Factor each epression completel. Check using multiplication..).).).).) m m.)
7 .).).) t 0t + 0.).) a a.) B. Solving quadratics using factoring To solve a quadratic equation is to find the 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 Propert: Zero Product Propert List some pairs of numbers that multipl to zero: ( )( ) = 0 ( )( ) = 0 ( )( ) = 0 ( )( ) = 0 What did ou notice? ZERO PRODUCT PROPERTY Algebra If the of two epressions is zero, then or of the epressions equals zero. If A and B are epressions and AB =, then A = or B =. Eample If ( + )( + ) = 0, then + = 0 or + = 0. That is, = or =. Use this pattern to solve for the variable:. get the quadratic = 0 and factor completel. set each ( ) = 0 (this means to write two new equations). solve for the variable (ou sometimes get more than solution)
8 Find the roots of each equation:.) 0 0.) 0.) Find the zeros of each equation:.) 0.) v(v + ) = 0.) Find the zeros of the function b rewriting the function in intercept form:.) 0.) f g.) Graph the function. Label the verte and ais of smmetr: ( ).) HW #: pg. #- s, pg. #- s, Pg. : #- Stud for Quiz on Sec..-. (Quiz is tomorrow!) Verte: Maimum or minimum value: -intercepts: Ais of smmetr: Compare width to the graph of =
9 Notes #: Section - Solve Quadratic Equations b Finding Square Roots A. Simplifing Square Roots: Make a factor tree; circle pairs of buddies. One of each pair comes out of the root, the non-paired numbers sta in the root. Multipl the terms on the outside together; multipl the terms on the inside together Simplif:.) 0.) B. Multipling Square Roots: Simplif each radical completel b taking out buddies (outside outside) inside inside or ( a b)( c d ) Simplif our answer, if possible Simplif:.).) C. Simplifing Square Roots in Fractions: Split up the fraction: a b Simplif first b taking out buddies or reducing (ou can onl reduce two numbers that are both under a root or two numbers that are both not in a root) Square root top, square root bottom If one square root is left in the denominator, multipl the top and the bottom b the square root and simplif OR If a binomial is left in the denominator, then multipl top and bottom b the conjugate of the denominator (eact same epression ecept with the opposite sign). Remember to FOIL on the denominator. Reduce if possible Simplif:.).) +.)
10 D. Solving Quadratic Equations Using Square Roots Isolate the variable or epression being squared (get it ) Square root both sides of the equation (include + and on the right side!) This means ou have equations to solve!! Solve for the variable (make sure there are no roots in the denominator).) =.) = 0.) = 0.) m.) ( + ) =.) HW #: Pg. 0: - even, Pg. : - odd, pg. #- s 0
11 Notes#: Sections - & - Comple Numbers and Completing the square Section.: Comple Numbers A. Definitions Define Comple Numbers: imaginar unit (i): imaginar number : B. Solving a quadratic equation with comple roots Isolate the epression being squared Square root both sides; write two equations Replace with i. Simplif Solve.) =.) + =.) ( ) 0 C. Adding, subtracting, and multipling comple numbers Distribute/FOIL. Combine like terms. Replace i with (-). Simplif. Simplif.) ( + i) ( i).) ( + i) + ( i).) ( + i)( + i).) ( i)( i) B. Dividing comple numbers If i is part of a monomial on the denominator, multipl top and bottom b i. e: i If i is part of a binomial on the denominator, multipl top and bottom b the comple conjugate of the denominator (same epression but opposite sign). FOIL. e: i Replace i with (-). Simplif..) i.) i i 0.) i i.) i i
12 Section.: 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; equations! Solve for the variable (make sure there are no roots in the denominator) ) (k + ) =.) + + =.) n n + = C. Completing the Square a b c 0 Take half the b (the coefficient) Square this number (no decimals leave as a fraction!) Add this number to the epression Factor it should be a binomial, squared ( ).) + +.) m m + ( )( ) ( ) Find the value of c such that each epression is a perfect square trinomial. Then write the epression as the square of a binomial..) w + w + c.) k k + c Solving b Completing the Square: Collect variables on the left, numbers on the right Divide ALL terms b 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 (simplif all roots look for i ).) + = 0.) m m + = 0
13 0.) k k k.) w w w.) = 0.) Cumulative Review: Solving Quadratics Solve b factoring:.) k k =.) m = 0 Solve b using square roots:.) w =.) = 0 HW #: pg. #- s pg. #- s pg. 0: - even
14 Notes #: Sec. - Use the Quadratic Formula and the Discriminant A. Review of Simplifing Radicals and Fractions Simplif epression under the radical sign ( i ); reduce Reduce onl 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 ou have TWO answers Simplif:.).) 0.) 0.).) ( ) ()()().) () ( )( ) B. Solving Quadratics using the Quadratic Formula So far, we have solved quadratics b: (), (), and (). The final method for solving quadratics is to use the quadratic formula. Solving using the quadratic formula: Put into standard form (a + b + c = 0) List a =, b =, c = Plug a, b, and c into b b ac a Simplif all roots (look for i ); reduce
15 Solve b using the quadratic formula:.) + = (std. form): a = b = c = b b ac a.) = - (std. form): a = b = c = b b ac a.) - + = -.) =.) - + =.) ( )
16 C. Using the Discriminant Quadratic equations can have two, one, or no solutions (-intercepts). You can determine how man solutions a quadratic equation has before ou solve it b using the. The discriminant is the epression under the radical in the quadratic formula: Discriminant = b ac If b ac < 0, then the equation has imaginar solutions If b ac = 0, then the equation has real solution If b ac > 0, then the equation has real solutions b b ac a A. Finding the number of -intercepts Determine whether the graphs intersect the -ais in zero, one, or two points..).) 0 B. Finding the number and tpe of solutions Find the discriminant of the quadratic equation and give the number and tpe of solutions of the equation..).).) =.) = +
17 Cumulative Review Problems: Solve b factoring:.) m +m = 0.) = 0 Solve b using square roots:.) b + = 0 0.) ( + ) = Solve b completing the square:.) m + m + = 0.) = 0 For #, find the verte of the parabola. Graph the function and find the requested information.) g() = Verte: Ma or min value: Direction of opening? Compare width to = : Ais of smmetr: HW #: Pg. #- pg. #- s pg. #-even
18 Notes #0: Section - Graph and Solve Quadratic Inequalities GRAPHING A QUADRATIC INEQUALITY IN TWO VARIABLES To graph a quadratic inequalit, follow these steps: Step Graph the parabola with equation = a + b + c. Make the parabola for inequalities with < or > and for inequalities with or. Step Test a point (, ) the parabola to determine whether the point is a solution of the inequalit. Step Shade the region the parabola if the point from Step is a solution. Shade the region the parabola if it is not a solution..) )
19 .) > ) HW #0: pg. 0 #- s Pg. #- all
20 Notes# : Sec. -0 Write Quadratic Functions and Models A. When given the verte and a point Plug the verte in for (h, k) in a( h) k Plug in the given point for (, ) Solve for a. Plug in a, h, k into a( h) k.) Write a quadratic equation in verte form for the parabola shown..) Write a quadratic function in verte form for the function whose graph has its verte at (, ) and passes through the point (, ). B. When given the -intercepts and a third point Plug in the -intercepts as p and q into = a( p)( q) Plug in the given point for (, ) Solve for a. Plug in a, h, k into = a( p)( q).) Write a quadratic function in intercept form for the parabola shown. 0
21 B. When given three points on the parabola Label all three points as (, ) Separatel, plug in each point into a b c You now have equations with three variables: a, b, c Solve for a, b, and c using elimination (see notes #). Plug back into a b c.) Write a quadratic function in standard form for the parabola that passes through the points (, ), (0, ) and (, )..) Write a quadratic function in standard form for the parabola that passes through the points (, ), (, ) and (, ). HW #: pg. #-even Pg. 0: - odd, - all,, Chapter Test and Notes Check on Frida
22 Notes #: Chapter Review To graph a quadratic function, ou must FIRST find the verte (h, k)!! (A) If the function starts in standard form a b c : st b : The -coordinate of the verte, h, = a nd : Find the -coordinate of the verte, k, b plugging the -coordinate into the function & solving for. (B) If the function starts in intercept form a( p)( q) : st : Find the -intercepts b setting the factors with equal to 0 & solving for. nd : The -coordinate of the verte is half wa between the -intercepts. rd : Find the -coordinate of the verte, k, b plugging the -coordinate into the function & solving for. (C) If the function starts in verte form a( h) k : st : pick out the -coordinate of the verte, h. REMEMBER: h will have the OPPOSITE sign as what is in the parenthesis!! nd : Pick out the -coordinate of the verte, k. It will have the SAME sign as the what is in the equation! AFTER finding the verte: Make a table of values with points: The verte, plug in -coordinates SMALLER than the -coordinate of the verte & -coordinates LARGER than the -coordinate of the verte. 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, the graph is narrower than If a, the graph is wider than
23 Graph each function b making a table of values with at least points. (A) State the verte. (B) State the direction of opening (up/down). (C) State whether the graph is wider, narrower, or the same width as..) f ( ) ( ) ) k() = + + Verte: Direction of opening? Compare width to the graph of = ) f() = Verte: Direction of opening? Compare width to the graph of = Verte: Direction of opening? Compare width to the graph of =
24 .) f ( ) ) h( ) Verte: Direction of opening? Compare width to the graph of = ) g( ) ( )( ) Verte: Direction of opening? Compare width to the graph of = Verte: Direction of opening? Compare width to the graph of =
25 Methods for Solving Quadratic Equations: A.) Factoring st : Set equal to 0 nd : Factor out the GCF rd : Complete the X & bo method to find the factors th : Set ever factor that contains an in it, equal to 0 & solve for. B.) Completing the Square st : 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 : Divide ever term in the equation b the value of a, if it is not alread. rd b : Create a perfect square trinomial on the left side b adding to both sides. th b : Factor the left side into a and simplif the value on the right side. th : Take the square root of both sides of the equation. REMINDER: Don t forget the th : Solve for C.) Finding Square Roots st : Isolate the term with the square. nd : Take the square root of both sides of the equation. REMINDER: Don t forget the rd : Solve for. D.) Quadratic Formula st : Set the equation equal to 0. nd : Find the values of a, b, and c & plug them into the Quadratic Formula: b b ac a rd : Simplif the radical as much as possible. th : If possible, simplif the numerator into integers. th : Divide. REMINDER: If ou have terms in the numerator (e: ), divide BOTH terms b the number in the denominator (the eample would result in ) Eamples: Solve each equation b the method stated. B Square Roots:.) ( + ) =.) (m ) = 0.) (r )
26 B Factoring:.) = 0.).) B Completing the Square:.) = 0.) + = 0.) = 0 B Quadratic Formula: 0.) + =.) = -.) = HW # Chapter Review Sheet all (check answers online!!) Chapter Test and Notes Check on Frida
27 Chapter Review Sheet Please complete each problem on a separate sheet of paper. Show all of our work and please use graph paper for all graphs. For questions -, solve b factoring For questions -, solve b finding square roots.. ( ) 0. For questions -, solve b completing the square For questions -0, solve b using the quadratic formula For questions -, simplif each epression.... For questions -, write each function in verte form, graph the function, and label the verte and ais of smmetr.... ( ) For question -, graph each inequalit.. ( ).. For questions 0-, write a quadratic function in verte form whose graph has the given verte and passes through the given point. 0. Verte (, ) and passes through (-, ). Verte (-, -) and passes through (, -) For questions -, write a quadratic function in standard form whose graph passes through the given points.. (, ), (0, -), (, ). (-, -), (, -), (, -) For questions -, write the epression as a comple number in standard form.. ( i) ( i). ( i)( i). ( i) ( i). i i. i i Chapter Test and Notes Check Tomorrow!
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