Notes Packet on Quadratic Functions and Factoring Graphing quadratic equations in standard form, vertex form, and intercept form. A. Intro to Graphs of Quadratic Equations:! = ax + bx + c A is a function that can be written in the form! = ax + bx + c where a, b, and c are real numbers and a! 0. The graph of a quadratic function is a U-shaped curve called a. The maximum or minimum point is called the Identif the vertex of each graph; identif whether it is a minimum or a maximum..).) Vertex: (, ) Vertex: (, ).).) Vertex: (, ) Vertex: (, ) B. Ke Features of a Parabola: Without graphing the quadratic functions, complete the requested information:.)! f ( x) = x x + What is the direction of opening? Is the vertex a max or min? Wider or narrower than = x? g( x) = x + x.)! What is the direction of opening? Is the vertex a max or min? Wider or narrower than = x? = x.)! What is the direction of opening? Is the vertex a max or min? Wider or narrower than = x?.)! = 0.x +.x. What is the direction of opening? Is the vertex a max or min? Wider or narrower than = x?
!!! The parabola = x 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 (! = ax + bx + c ): we need to find the vertex first. Find the vertex of each parabola. Graph the function and find the requested information.) f(x)= -x + x + a =, b =, c = 0-0 - - - - - - 0 x - - - - - - -0 Vertex: Max or min? Direction of opening? Axis of smmetr: Compare to the graph of = x 0.) h(x) = x + x + 0-0 - - - - - - 0 x - - - - - - -0 Vertex: Max or min? Direction of opening? Axis of smmetr: Compare to the graph of = x
!.) k(x) = x x 0-0 - - - - - - 0 x - - - - - - -0 Vertex: Max or min? Direction of opening? Axis of smmetr: Compare to the graph of = x.) State whether the function = x + x has a minimum value or a maximum value. Then find the minimum or maximum value. = x + x.) Find the vertex of!. State whether it is a minimum or maximum. Find that minimum or maximum value.
!! Another useful form of the quadratic function is the vertex form:. GRAPH OF VERTEX FORM = a(x h) + k The graph of = a(x h) + k is the parabola = ax translated h units and k units. The vertex is (, ). The axis of smmetr is x =. The graph opens up if a 0 and down if a 0. Find the vertex of each parabola and graph. ( ) = x +.)! 0-0 - - - - - - 0 x - - - - - - -0 = ( x ).)! + 0-0 - - - - - - 0 x - - - - - - -0 Vertex: Vertex:.) Write a quadratic function in vertex form for the function whose graph has its vertex at (-, ) and passes through the point (, ).
! GRAPH OF INTERCEPT FORM = a(x p)(x q): Characteristics of the graph = a(x p)(x q): The x-intercepts are and. The axis of smmetr is halfwa between (, 0) and (, 0) and it has equation x =! The graph opens up if a 0 and opens down if a 0..) Graph = (x )(x ) 0-0 - - - - - - 0 x - - - - - - -0 Converting between forms: From intercept form to standard form Use FOIL to multipl the binomials together Distribute the coefficient to all terms x-intercepts:, Vertex: ( )( ) = x + x Ex:! From vertex 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 ( ) ( ) f x = x + Ex:!
Notes : Solving quadratics b Factoring A. Factoring Quadratics Examples of monomials: Examples of binomials: Examples 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-Box method () Check our factoring b using multiplication/foil! Factor each expression completel. Check using multiplication..)! x x.)! x.)! x x.)! x.)! m m +.)! x + x +
.)! x x +.)! x + x.) t 0t + 0.)! x.)! a + a +.)! x + x + 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 Propert: Zero Product Propert List some pairs of numbers that multipl to zero: ( )( ) = 0 ( )( ) = 0 ( )( ) = 0 ( )( ) = 0 What did ou 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 + )(x + ) = 0, then x + = 0 or x + = 0. That is, x = or x =.
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) Find the roots of each equation:.)! x + x 0 = 0 x x + = 0.)!.) x x = Find the zeros of each equation:.)! x + x 0 = x.) v(v + ) = 0.) x + x = Find the zeros of the function b rewriting the function in intercept form:.)! = x x f x = x + x 0.)! ( ) g x.)! ( ) x =
Graph the function. Label the vertex and axis of smmetr: ( ) = x ( x + ).)! 0-0 - - - - - - 0 x - - - - - - -0 Vertex: Maximum or minimum value: x-intercepts: Axis of smmetr: Compare width to the graph of = x
Notes #: 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: a Split up the fraction:! 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 (exact same expression except with the opposite sign). Remember to FOIL on the denominator. Reduce if possible Simplif:.)!.)! +.)! +
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 ou have equations to solve!! Solve for the variable (make sure there are no roots in the denominator).) x =.) x = 0.) x = 0.) m =.) ( + ) =.) ( x ) =
Notes#: Complex Numbers and Completing the square Complex Numbers A. Definitions Define Complex Numbers: imaginar unit (i): imaginar number : B. Solving a quadratic equation with complex roots Isolate the expression being squared Square root both sides; write two equations Replace! with i. Simplif Solve.) x =.) x + =.)! (x ) + = 0 C. Adding, subtracting, and multipling complex numbers Distribute/FOIL. Combine like terms. Replace! i with (-). Simplif. Simplif.) ( + i) ( i).) ( + i) + ( i).) ( + i)( + i).) ( i)( i)
! B. Dividing complex numbers If i is part of a monomial on the denominator, multipl top and bottom b i. ex: i If i is part of a binomial on the denominator, multipl top and bottom b the complex conjugate of the denominator (same expression but opposite sign). FOIL. ex:! i Replace! i with (-). Simplif.!!!! + i i + i.) i.) + i 0.) i.) i
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 + ) =.) x + x + =.) n n + = C. Completing the Square! ax + 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 ( ).) x + x +.) m m + ( )( ) ( ) Find the value of c such that each expression is a perfect square trinomial. Then write the expression 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 ).) x + x = 0.) m m + = 0
0.)! k + k = k.)! w + w + = w.) x x = 0.) x + x = x Cumulative Review: Solving Quadratics Solve b factoring:.) k k =.) m = 0 Solve b using square roots:.) w =.) = 0
Notes #: Use the Quadratic Formula and the Discriminant A. Review of Simplifing Radicals and Fractions Simplif expression 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 (ax + bx + c = 0) List a =, b =, c = ± x = b b ac a Plug a, b, and c into! Simplif all roots (look for! = i ); reduce.) x + x = Solve b using the quadratic formula: ± x = b b ac a (std. form): a = b = c =.) x x = - ± x = b b ac a (std. form): a = b = c =
.) -x + x = -.) x = x.) -x + x =.) ( x ) = x +
! C. Using the Discriminant Quadratic equations can have two, one, or no solutions (x-intercepts). You can determine how man solutions a quadratic equation has before ou solve it b using the. The discriminant is the expression under the radical in the quadratic formula: ± x = b b ac a 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 A. Finding the number of x-intercepts Determine whether the graphs intersect the x-axis in zero, one, or two points..)! = x x +.)! = x x 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..)! x x =.)! x = x.) x x =.) x = x +
! Cumulative Review Problems: Solve b factoring:.) m +m = 0.) x x = 0 Solve b using square roots:.) b + = 0 0.) (x + ) = Solve b completing the square:.) m + m + = 0.) x x = 0 For #, find the vertex of the parabola. Graph the function and find the requested information.) g(x) = -x + x 0-0 - - - - - - 0 x - - - - - - -0 Vertex: Max or min value: Direction of opening? Compare width to = x : Axis of smmetr:
!! Notes #0: 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 = ax + bx + c. Make the parabola for inequalities with < or > and for inequalities with or. Step Test a point (x, ) 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..) x + x +.) x + x 0-0 - - - - - - 0 x - - - - - - -0 0-0 - - - - - - 0 x - - - - - - -0
!!.) > x +x < x +x + 0 0-0 - - - - - - 0 x - - - - - - -0.) < -x + > x x 0-0 - - - - - - 0 x - - - - - - -0
Notes# : Write Quadratic Functions and Models A. When given the vertex and a point Plug the vertex in for (h, k) in! = a( x h) + k Plug in the given point for (x, ) Solve for a. Plug in a, h, k into! = a( x h) + k.) Write a quadratic equation in vertex form for the parabola shown..) Write a quadratic function in vertex form for the function whose graph has its vertex at (, ) and passes through the point (, ). B. When given the x-intercepts and a third point Plug in the x-intercepts as p and q into = a(x p)(x q) Plug in the given point for (x, ) Solve for a. Plug in a, h, k into = a(x p)(x q).) Write a quadratic function in intercept form for the parabola shown.
! B. When given three points on the parabola Label all three points as (x, ) Separatel, plug in each point into! = ax + bx + c You now have equations with three variables: a, b, c Solve for a, b, and c using elimination (see notes #). Plug back into = ax + bx + 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 (, ).
Notes #: Review To graph a quadratic function, ou must FIRST find the vertex (h, k)!! (A) If the function starts in standard form! = ax + bx + c : b st : The x-coordinate of the vertex, h, = a nd : Find the -coordinate of the vertex, k, b plugging the x-coordinate into the function & solving for. (B) If the function starts in intercept form! = a( x p)( x q) : st : Find the x-intercepts b setting the factors with x equal to 0 & solving for x. nd : The x-coordinate of the vertex is half wa between the x-intercepts. rd : Find the -coordinate of the vertex, k, b plugging the x-coordinate into the function & solving for. (C) If the function starts in vertex form! = a( x h) + k : st : pick out the x-coordinate of the vertex, h. REMEMBER: h will have the OPPOSITE sign as what is in the parenthesis!! nd : Pick out the -coordinate of the vertex, k. It will have the SAME sign as the what is in the equation! AFTER finding the vertex: Make a table of values with points: The vertex, plug in x-coordinates SMALLER than the x-coordinate of the vertex & x-coordinates LARGER than the x-coordinate of the vertex. Direction of Opening: If a is positive, the graph opens up If a is negative, the graph opens down. Width of the function: a > If!, the graph is narrower than! = x a < If!, the graph is wider than! = x
!!! Graph each function b making a table of values with at least points. (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! = x..)! f ( x) = ( x + ) 0-0 - - - - - - 0 x - - - - - - -0 Vertex: Direction of opening? Compare width to the graph of = x.) k(x) = x + x + 0-0 - - - - - - 0 x - - - - - - -0 Vertex: Direction of opening? Compare width to the graph of = x.) f(x) = x x 0-0 - - - - - - 0 x - - - - - - -0 Vertex: Direction of opening? Compare width to the graph of = x
!!!.)! f ( x) = x + x + 0-0 - - - - - - 0 x - - - - - - -0 Vertex: Direction of opening? Compare width to the graph of = x.)! h( x) = x + x + 0-0 - - - - - - 0 x - - - - - - -0 g( x) = ( x )( x + ).)! 0-0 - - - - - - 0 x - - - - - - -0 Vertex: Direction of opening? Compare width to the graph of = x Vertex: Direction of opening? Compare width to the graph of = x
! Methods for Solving Quadratic Equations: A.) Factoring st : Set equal to 0 nd : Factor out the GCF rd : Complete the X & box method to find the factors th : Set ever factor that contains an x in it, equal to 0 & solve for x. 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. b rd : Create a perfect square trinomial on the left side b adding to both sides. b x ± th : 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 x 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 x. 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 x = 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 (ex: ), divide BOTH terms b the number in the denominator (the example would result in ± ) Examples: Solve each equation b the method stated. B Square Roots: ±.) ( + ) =.) (m ) = 0.)! (r + ) =
B Factoring:.) x x = 0.) x x =.) x + x = B Completing the Square:.) x x = 0.) + = 0.) x x = 0 B Quadratic Formula: 0.) x + x =.) x x = -.) x = x
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..! x x + = 0.! x x + = 0.! x + x 0 = 0 For questions -, solve b finding square roots. ( ) 0.! x + = ( x ) + =.! For questions, solve b completing the square. 0.! x x + = 0.! x + 0x + 0 = 0.! x + x = For questions -0, solve b using the quadratic formula..! x x = 0 0.! x + x = x + x For questions -, simplif each expression. ±.!.!.! For questions, write each function in vertex form, graph the function, and label the vertex and axis of smmetr. = ( x ).! = x x +.! = x x +.! For question, graph each inequalit..! > ( x + ).! x + x +.! x x + For questions 0-, write a quadratic function in vertex form whose graph has the given vertex and passes through the given point. 0. Vertex (, ) and passes through (-, ). Vertex (-, -) and passes through (, -) For questions -, write a quadratic function in standard form whose graph passes through the given points.. (, ), (0, -), (, ). (-, -), (, -), (, -) For questions, write the expression as a complex number in standard form..! ( i) + ( + i).! ( i)( + i).! ( i) ( i)!! + i i. i. i