Chapter Summar Ke Terms standard form of a quadratic function (.1) factored form of a quadratic function (.1) verte form of a quadratic function (.1) concavit of a parabola (.1) reference points (.) transformation (.) rigid motion (.) argument of a function (.) translation (.) vertical dilation (.) vertical stretching (.) vertical compression (.) reflection (.) line of reflection (.) horizontal dilation (.) horizontal stretching (.) horizontal compression (.) the imaginar number i (.6) principal square root of a negative number (.6) set of imaginar numbers (.6) pure imaginar number (.6) set of comple numbers (.6) real part of a comple number (.6) imaginar part of a comple number (.6) comple conjugates (.6) monomial (.6) binomial (.6) trinomial (.6) imaginar roots (.7) discriminant (.7) imaginar zeros (.7) Fundamental Theorem of Algebra (.7) double root (.7).1 Using Characteristics of a Quadratic Function to Describe Its Graph The graphs of quadratic functions can be described using ke characteristics: -intercept(s) -intercept verte ais of smmetr concave up or down The -intercept (c) and whether the parabola is concave up (a. 0) or down (a, 0) can be determined when the quadratic is in standard form, f() 5 a 1 b 1 c. The -intercepts (r 1, 0), (r, 0), and whether the parabola is concave up (a. 0) or down (a, 0) can be determined when the quadratic is in factored form, f() 5 a( r 1 )( r ). The verte (h, k), whether the parabola is concave up (a. 0) or down (a, 0), and the ais of smmetr ( 5 h) can be determined when the quadratic is in verte form, f() 5 a( h) 1 k. The eample shows that using the ke characteristics of a graph can also determine the function represented b the graph. 79
Analze the graph. Then, circle the function(s) which could model the graph. Describe the reasoning ou used to either eliminate or choose each function. f 1 () 5 ( )( 1 ) The function f 1 can be eliminated because the a-value is positive. f () 5 1 ( 1 )( ) The function f can be eliminated because the a-value is positive. f () 5 1 ( 1 )( ) The function f is a possible function because it has a negative a-value and one positive and one negative -intercept. f () 5 1 ( )( ) The function f can be eliminated because it has two positive -intercepts..1 Determining the Best Form in Which to Write a Quadratic Function Given the roots or -intercepts and an additional point, a quadratic function can be written in factored form: f() 5 a( r 1 )( r ). Given a minimum or maimum point or verte and an additional point, a quadratic function can be written in verte form: f() 5 a( h) 1 k. Given a -intercept and two additional points, a quadratic function can be written in standard form: f() 5 a 1 b 1 c. verte (, 5) and point (, 1): verte form points (, 0), (6, 5), and (9, 0): factored form -intercept (0, ) and points (, ), (5, ): standard form.1 Writing a Quadratic Function in Standard Form When a quadratic function is in verte or factored form, the function can be rewritten in standard form b multipling the factors and combining like terms. f() 5 ( )( 1 8) f() 5 ( ) 1 6 f() 5 ( )( 1 8) f() 5 ( ) 1 6 5 1 8 16 5 ( 6 1 9) 1 6 5 1 6 16 5 1 1 18 1 6 5 1 1 1 80 Chapter Quadratic Functions
.1 Writing a Quadratic Function to Represent a Situation Determine the important information from the problem situation, including a possible verte or -intercepts. Substitute the information into either the verte or factored form of a quadratic function and solve for a. Write the complete function. Hector launches a rocket from a platform 5 feet above the ground. The rocket reaches a maimum height of 50 feet at a distance of 75 feet. Write a function to represent the rocket s height in terms of its distance. h(d) 5 a(d 50) 1 75 5 5 a(0 50) 1 75 5 5 500a 1 75 70 5 500a 7 50 5 a h(d) 5 7 50 (d 50) 1 75. Identifing the Reference Points of a Quadratic Function Reference points are a set of ke points that help identif a basic function. Reference Points of Basic Quadratic Function P (0, 0) Q (1, 1) R (, ) If the verte of a quadratic function and two points to the right of that verte are known, the ais of smmetr can be used to draw the other half of the parabola. These ke points are reference points for the quadratic function famil. R(, ) Q(1, 1) P(0, 0) Chapter Summar 81
. Understanding the Effect of the C-Value and the D-Value in the Transformational Function Form g() 5 Af(B( C)) 1 D A translation shifts an entire graph the same distance and direction. When f() is transformed to g() 5 Af(B( C)) 1 D, the C-value shifts the graph horizontall and the D-value shifts the graph verticall. When C, 0, the graph shifts left C units, and when C. 0, the graph shifts to the right C units. When D, 0, the graph shifts down D units, and when D. 0, the graph shifts up D units. f() 5 m() 5 ( 1 ) 1 1 C 5 and D 5 1 The original function f() will translate to the left units and translate up 1 unit. Reference Points on f() Appl the Transformations Corresponding Points on m() (0, 0) (0, 0 1 1) (, 1) m() 8 6 8 6 0 6 8 (1, 1) (1, 1 1 1) (, ) 6 (, ) (, 1 1) (1, 5) 8. Writing the Function of a Transformed Graph Identif the location of the new verte. Use the - and -coordinates of the new verte as the C- and D-values. Write the transformed function in terms of the original function in the form g() 5 f( C) 1 D. Given 5 f() g() 5 f( ) f() g() 1 1 0 1 5 1 8 Chapter Quadratic Functions
. Understanding the Effect of the A-Value in the Transformational Function Form g() 5 Af(B( C)) 1 D A translation shifts an entire graph the same distance and direction. When f() is transformed to g() 5 Af(B( C)) 1 D, the A-value stretches or compresses the graph verticall. The C-value shifts the graph horizontall and the D-value shifts the graph verticall. When A $ 1, the graph stretches awa from the -ais, and when 0, A, 1, the graph compresses towards the -ais. When A, 0, the graph is reflected across the -ais or across the horizontal line 5 D. An point on the transformed graph can be represented in coordinate notation as (, ) ( 1 B 1 C, A 1 D ). f() 5 b() 5 (f( 1 1) The C-value is 1 and the D- value is so the verte will be translated 1 unit to the left and units down to (1, ). The A-value is, so the graph will have a vertical stretch b a factor of and will be reflected across the line 5. (, ) 5 ( 1, ). Writing a Quadratic Function to Represent a Graph Given 5 f(), the coordinate notation represented in the transformational function 5 Af( C) 1 D is (, ) becomes ( 1 C, A 1 D). Using this coordinate notation and reference points, ou can work backwards from the graph to write the function in verte form. 8 6 8 6 0 6 8 6 8 f() 5 1 ( ) Chapter Summar 8
. Graphing Horizontal Dilations of Quadratic Functions Considering the transformational function g() 5 Af(B( C)) 1 D, B determines the horizontal dilation. Horizontal stretching is the stretching of a graph awa from the -ais and happens when 0, B, 1. Horizontal compression is the squeezing of a graph towards the -ais and happens when B. 1. The factor of horizontal stretch or compression is the reciprocal of the B-value. When B, 0, the dilation will also include a reflection across the -ais, but because a parabola is smmetric, it will look identical to the function when B. 0. p() 5 f() Reference Points on f() (0, 0) ( 1 (1, 1) ( 1 (, ) ( 1 (, 9) ( 1 Appl the Transformations? 0, 0 )? 1, 1 )?, )?, 9 ) Corresponding Points on p() (0, 0) (0.5, 1) (1, ) (1.5, 9) 16 1 1 10 8 6 1 0 1 8 Chapter Quadratic Functions
. Identifing Multiple Transformations Given Quadratic Functions Analzing the transformations of 5 f(), it is possible to graph a function from information given b the form of the equation. Function Form Equation Information Description of Transformation of Graph D. 0 D, 0 C. 0 C, 0 vertical shift up D units vertical shift down D units horizontal shift right C units horizontal shift left C units 5 Af (B( C)) 1 D A. 1 vertical stretch b a factor of A units 0, A, 1 vertical compression b a factor of A units A, 0 reflection across the line 5 k B. 1 horizontal compression b a factor of 1 B 0, B, 1 horizontal stretch b a factor of 1 B B, 0 reflection across the -ais h() 5 ( 1 7) 8 The A value is so the graph will have a vertical stretch b a factor of and be reflected over the -ais. The C-value is 7 and the D-value is 8, so the verte will be shifted 7 units to the left and 8 units down to (7, 8). The A-value is so the graph will have a vertical stretch b a factor of and be reflected across the line 5 D, or 5 8 Given the graph of 5 f(), the graph of g() 5 f ( 1 ( ) ) 5 is sketched. 6 8 g() 6 0 6 8 10 6 8 f() Chapter Summar 85
. Writing Quadratic Functions in Terms of a Given a Graph Determine b how much and in what direction the reference points of the graph have moved and if there has been a reflection, horizontal dilation, or vertical dilation. Use coordinate notation ( 1 B 1 C, A 1 D ) to help identif A, B, C, and D values. Write the transformed function in terms of the original function in the form g() 5 Af(B( C)) 1 D. The function of g() in terms of f() is g() 5 f() 5, or g() 5 f( 1) 5. 8 6 8 f() 6 0 6 8 g() 6 8.5 Deriving a Quadratic Equation Given Certain Information and Reference Points To write a unique quadratic function, use the reference points of the basic quadratic function and consider the vertical distance between each point. If given the -intercepts and another point on the parabola, a function can be written in factored form or if given the verte and another point on the parabola, a function can be written in verte form; but in both cases the a-value must also be determined. To do this, first determine the ais of smmetr as directl in the middle of the two -intercepts or at the location of the verte. Plot the ais of smmetr and points on a coordinate plane and assign each point a letter based on its horizontal distance from the line of smmetr. The vertical distance between A9 and B9 will be the vertical distance between A and B of the basic function times a. -intercepts and 1 point: (, 0), (5, 0), (, 1) r 1 r 1 5 1 5 5 5 1 so the ais of smmetr is 5 1. Point (5, 0) is point D9 because it is units from the ais of smmetr. Point (, 1) is point C9 because it is units from the ais of smmetr. The range between the C and D point on the basic function is 7. The range between point D9 and point C9 is 7 () 5 1, therefore the a-value must be. f() 5 ( 1 ) ( 5) (, 0) 16 1 1 10 8 6 units C9 (, 1) 1 0 1 units 1 (5, 0) D9 86 Chapter Quadratic Functions
.5 Deriving a Quadratic Equation Given Three Points Using a Graphing Calculator You can use a graphing calculator to determine a quadratic regression equation given three points on the parabola. Step 1: Diagnostics must be turned on so that all needed data is displaed. Press nd CATALOG to displa the catalog. Scroll to DiagnosticOn and press ENTER. Then press ENTER again. The calculator should displa the word Done. Step : Press STAT and then press ENTER to select 1:Edit. In the L1 column, enter the -values b tping each value followed b ENTER. Use the right arrow ke to move to the L column. Enter the -values. Step : Press STAT and use the right arrow ke to show the CALC menu. Choose 5:QuadReg. Press Enter. The values for a, b, and c will be displaed. Step : To have the calculator graph the eact equation for ou, press Y=, VARS, 5:Statistics, scroll right to EQ, press 1:RegEQ, GRAPH. (8, 1), (1, 1.5), (6, 8) f() 5 0.75 5 1 9 Chapter Summar 87
.5 Deriving a Quadratic Equation Given Three Points Using a Sstem of Equations To use a sstem of equations, first create a quadratic equation in the form 5 a 1 b 1 c for each of the points given. Then, use elimination and substitution to solve for a, b, and c. (1, 11), (, 5), (, 1) Equation 1: 11 5 a(1) 1 b(1) 1 c Equation : 5 5 a() 1 b() 1 c 11 5 a b 1 c 5 5 a 1 b 1 c Equation : 1 5 a() 1 b() 1 c 1 5 16a 1 b 1 c Subtract Equation 1 from Equation and solve in terms of a: 5 5 a 1 b 1 c (11 5 a b 1 c) 6 5 a 1 b b 5 a Subtract Equation from Equation : 1 5 16a 1 b 1 c (5 5 a 1 b 1 c) 6 5 1a 1 b 18 5 6a 1 b Substitute the value for a into this equation: 18 5 6( b) 1 b 18 5 1 5b 0 5 5b b 5 6 Substitute the value of b into the equation for the value of a: a 5 6 a 5 Substitute the values of a and b into Equation 1: 11 5 6 1 c 11 5 10 1 c 1 5 c Substitute the values for a, b, and c into a quadratic equation in standard form: f() 5 1 6 1 88 Chapter Quadratic Functions
.6 Calculating Powers of i The imaginar number i is a number such that i 5 1. Because no real number eists such that its square is equal to a negative number, the number i is not a part of the real number sstem. The values of i n repeat after ever four powers of i, where i 5 1, i 5 1, i 5 1, and i 5 1. i 5 5 (i ) 6 (i 1 ) 5 (1) 6 ( 1 ) 5 1.6 Rewriting Epressions with Negative Roots Using i Epressions with negative roots can be rewritten. For an positive real number n, the principal square root of a negative number, n, is defined b n 5 i n. 6 1 5 i 6 1 i 5 i (9)(7) 1 i (6)() 5 7 i 1 6 i.6 Adding, Subtracting, and Multipling on the Set of Comple Numbers The set of comple numbers is the set of all numbers written in the form a 1 bi, where a and b are real numbers. The term a is called the real part of a comple number, and the term bi is called the imaginar part of a comple number. When operating with comple numbers involving i, combine like terms b treating i as a variable (even though it s a constant). Comple conjugates are pairs of numbers of the form a 1 bi and a bi. The product of a pair of comple conjugates is alwas a real number in the form a 1 b. 1 ( 1 i )( i ) 5 1 7 1 ( 1 i )( i ) 5 1 7 5 1 9i 5 1 7 5 1 ( 5) 1 (7 9(1)) 5 1 16 Chapter Summar 89
.6 Identifing Comple Polnomials A polnomial is a mathematical epression involving the sum of powers in one or more variables multiplied b coefficients. The definition of a polnomial can be etended to include imaginar numbers. Some polnomials have special names, according to the number of terms the have. A polnomial with one term is called a monomial. A polnomial with two terms is called a binomial. A polnomial with three terms is called a trinomial. Combine like terms to name the polnomial. 5 i 1 i 1 9 The epression is a trinomial because it can be rewritten as 1 (5 i ) 1 (9 i ), which shows one term, one term, and one constant term..6 Adding, Subtracting, and Multipling Comple Polnomials Some polnomial epressions involving i can be simplified using methods similar to those used to operate with numerical epressions involving i. Whenever possible, multipl the comple conjugates first to get a real number. ( 1 7i )( 1 5i ) ( 1 7i )( 1 5i ) 5 6 1 15i 1 1i 1 5i 5 6 1 9i 1 5(1) 5 6 1 9i 5.6 Rewriting the Quotient of Comple Numbers When rewriting the quotient of comple numbers, multipl both the divisor and the dividend b the comple conjugate of the divisor, thus changing the divisor into a real number. The product of a pair of comple conjugates is alwas a real number in the form a 1 b. i 5 i 5 1 i? 5 i 5 1 i 5 i 15 6i 5i 1 i 5 5 1 5 15 11i 5 1 5 1 11i 9 5 1 11 9 9 i 90 Chapter Quadratic Functions
.7 Using the Quadratic Formula to Determine the Zeros of a Function If the graph of f() intersects the -ais times, then the quadratic equation f() 5 0 has solutions, or roots. If the graph of f() intersects the -ais 1 time, then the quadratic equation f() 5 0 has 1 solution and double roots. If the graph of f() does not intersect the -ais, then the quadratic equation f() 5 0 has no solution, or imaginar roots. Use the Quadratic Formula and what ou know about imaginar numbers to solve an equation of the form f() 5 0 to calculate the roots, or zeros of the function. Remember the Quadratic Formula is: 5 b 6 b ac. a g() 5 7 1 1 7 1 1 5 0 a 5 1, b 5 7, c 5 1 5 b 6 b ac a 5 (7) 6 (7) (1)(1) (1) 5 7 6 1 5, 5.7 Determining Whether a Quadratic Function in Standard Form Has Real or Imaginar Zeros The radicand in the Quadratic Formula, b ac, is called the discriminant because it discriminates the number and tpe of roots of a quadratic equation. If the discriminant is greater than or equal to zero, the quadratic equation has two real roots. If the discriminant is negative, the quadratic equation has two imaginar roots. If the discriminant is equal to zero, the quadratic equation appears to have onl 1 root, but still has real roots called a double root. The Fundamental Theorem of Algebra states that for an polnomial equation of degree n must have eactl n comple roots or solutions; also, ever polnomial function of degree n must have eactl n comple zeros. Zeros of a function f() are the values of for which f() 5 0 and are related to the roots of an equation. f() 5 1 1 b ac 5 () (1)(1) 5 9 56 5 7 The discriminant is negative, so the function has two imaginar zeros. Chapter Summar 91
.7 Determining Whether a Quadratic Function in Verte Form Has Real or Imaginar Zeros If the graph of f() intersects the -ais times, then the quadratic equation f() 5 0 has solutions, or roots. If the graph of f() intersects the -ais 1 time, then the quadratic equation f() 5 0 has 1 solutions and double roots. If the graph of f() does not intersect the -ais, then the quadratic equation f() 5 0 has no solution, or imaginar roots. So, b using C and D of verte form to locate the verte above or below the -ais and using A to determine if the parabola is concave up or down, ou can tell if the graph intersects the -ais or not..7 f() 5 5( ) 1 7 Because the verte (, 7) is above the -ais and the parabola is concave down (a, 0), it intersects the -ais. So, the zeros are real. Determining Whether a Quadratic Function in Factored Form Has Real or Imaginar Zeros Some functions can be factored over the set of real numbers, while other functions can be factored over the set of imaginar numbers. However, all functions can be factored over the set of comple numbers. If necessar, use the Quadratic Formula to determine the roots and write the function in factored form. p() 5 1 6 1 18 p() 5 [ ( 1 i )][ ( i )] 5 1 i, 5 i The function p() has two imaginar zeros. 9 Chapter Quadratic Functions