Answers (Anticipation Guide and Lesson 5-1)

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1 Chapter A1 Glencoe Algebra Step 1 Read each statement. Before ou begin Chapter Decide whether ou Agree (A) or Disagree (D) with the statement. Write A or D in the first column R if ou are not sure whether ou agree or disagree, write NS (Not Sure). STEP 1 A, D, or NS Step Anticipation Guide Quadratic Functions and Relations Statement 1. All quadratic functions have a term with the variable to the second power.. If the graph of the quadratic function = a + c opens up then c < 0.. A quadratic equation whose graph does not intersect the -ais has no real solution.. Since graphing shows the eact solutions to a quadratic equation, no other method is necessar for solving.. If ( - )( + ) = 0, then either - = 0 or + = 0.. An imaginar number contains i, which equals the square root of A method called completing the square can be used to rewrite a quadratic epression as a perfect square.. The quadratic formula can onl be used for quadratic equations that cannot be solved b graphing or completing the square. 9. The discriminant of a quadratic equation can be used to determine the direction the graph will open. 10. The graph of = is a dilation of the graph of =. 11. The graph of = ( + ) will be two units to the right of the graph of =. 1. The graph of a quadratic inequalit containing the smbol < will be a parabola opening downward. After ou complete Chapter Reread each statement and complete the last column b entering an A or a D. Did an of our opinions about the statements change from the first column? STEP A or D For those statements that ou mark with a D, use a piece of paper to write an eample of wh ou disagree. Chapter Glencoe Algebra A D A D A A A D D A D D Chapter Resources -1 Stud Guide and Intervention Graphing Quadratic Functions Graph Quadratic Functions Quadratic Function A function defined b an equation of the form f() = a + b + c, where a 0 Graph of a Quadratic Function A parabola with these characteristics: -intercept: c; ais of smmetr: = -b a ; - coordinate of verte: -b a Eample Find the -intercept, the equation of the ais of smmetr, and the -coordinate of the verte for the graph of f() = - +. Use this information to graph the function. a = 1, b = -, and c =, so the -intercept is. The equation of the ais of smmetr is = -(-) or (1). The -coordinate of the verte is. Net make a table of values for near. - + f() (, f()) (0) + (0, ) (1) + (1, ) ( ) - ( + ) 11 (, 11 ) - () + (, ) - () + (, ) Eercises Complete parts a c for each quadratic function. a. Find the -intercept, the equation of the ais of smmetr, and the -coordinate of the verte. b. Make a table of values that includes the verte. c. Use this information to graph the function. 1. f() = + +. f() = f() = - +, = -, -, = -1, -1, = 1, f() (-, -1) - f() 1 (-1, ) - - f() (1, 1) - Chapter Glencoe Algebra f() f() f() f() Lesson -1 Answers (Anticipation Guide and Lesson -1)

2 Chapter A Glencoe Algebra -1 Stud Guide and Intervention (continued) Graphing Quadratic Functions Maimum and Minimum Values The -coordinate of the verte of a quadratic function is the maimum value or minimum value of the function. Maimum or Minimum Value of a Quadratic Function The graph of f() = a + b + c, where a 0, opens up and has a minimum when a > 0. The graph opens down and has a maimum when a < 0. Eample Determine whether each function has a maimum or minimum value, and find that value. Then state the domain and range of the function. a. f() = For this function, a = and b = -. Since a > 0, the graph opens up, and the function has a minimum value. The minimum value is the -coordinate of the verte. The -coordinate of the verte is -b a = - () = 1. Evaluate the function at = 1 to find the minimum value. f(1) = (1) - (1) + 7 =, so the minimum value of the function is. The domain is all real numbers. The range is all reals greater than or equal to the minimum value, that is {f() f() }. b. f() = For this function, a = -1 and b = -. Since a < 0, the graph opens down, and the function has a maimum value. The maimum value is the -coordinate of the verte. The -coordinate of the verte is -b a = - - (-1) = -1. Evaluate the function at = -1 to find the maimum value. f(-1) = (-1) - (-1) = 101, so the minimum value of the function is 101. The domain is all real numbers. The range is all reals less than or equal to the maimum value, that is {f() f() 101}. Eercises Determine whether each function has a maimum or minimum value, and find that value. Then state the domain and range of the function. 1. f() = f() = f() = min., 9 7 ; all reals; min., -11; all reals; min., 1 ; all reals; { f() f() 9 7 } {f() f() -11} {f() f() 1 }. f() = + +. f() = f() = + + min., - 17 ; all reals; ma., 9; all reals; min., 1 ; all reals; 1 {f() f() - 17 } {f() f() 9} {f() f() } 7. f() = f() = f() = ma., 1; all reals; min., -0; all reals; ma., 7; all reals; {f() f() 1} {f() f() -0} {f() f() 7} Chapter Glencoe Algebra -1 Skills Practice Graphing Quadratic Functions Complete parts a c for each quadratic function. a. Find the -intercept, the equation of the ais of smmetr, and the -coordinate of the verte. b. Make a table of values that includes the verte. c. Use this information to graph the function. 1. f() = -. f() = - +. f() = - + 0; = 0; 0 ; = ; ; = ; f() f() (0, 0) f() f() Determine whether each function has a maimum or a minimum value, and find that value. Then state the domain and range of the function. (, 0) 0 f() f(). f() =. f() = -. f() = + min.; 0; ma.; 0; min.; -1; D = {all real numbers}; D = {all real numbers}; D = {all real numbers}; R = {f() f() 0} R = {f() f() 0} R = {f() f() -1} 7. f() = f() = f() = ma.; -1; min.; -9; min.; -1; D = {all real numbers}; D = {all real numbers}; D = {all real numbers}; R = {f() f() -1} R = {f() f() -9} R = {f() f() -1} 10. f() = 11. f() = f() = min.; 0; min.; 1; ma.; -; D = {all real numbers}; D = {all real numbers}; D = {all real numbers}; R = {f() f() 0} R = {f() f() 1} R = {f() f() -} 1. f() = f() = f() = min.; -11; min.; -0; ma.; 1; D = {all real numbers}; D = {all real numbers}; D = {all real numbers}; R = {f() f() -11} R = {f() f() -0} R = {f() f() 1} Chapter 7 Glencoe Algebra (, 1) Lesson -1 Answers (Lesson -1)

3 Chapter A Glencoe Algebra -1 Practice Graphing Quadratic Functions Complete parts a c for each quadratic function. a. Find the -intercept, the equation of the ais of smmetr, and the -coordinate of the verte. b. Make a table of values that includes the verte. c. Use this information to graph the function. 1. f() = f() = f() = ; = ; 1; = -; - 1; = 0.; 0. 0 f() f() 1 1 (, -1) f() (-, 1) - - Chapter Glencoe Algebra 1 1 f() f() f() (0., 0.) - - Determine whether each function has a maimum or minimum value, and find that value. Then state the domain and range of the function.. f() = + -. f() = v() = min.; -9; all reals; min.; ; all reals; ma.; -; all reals; {f() f() -9} {f() f() } {f() f() -} 7. f() = + -. f() = f() = min.; -; all reals; ma.; ; all reals; ma.; 0; all reals; {f() f() -} {f() f() } {f() f() 0} 10. GRAVITATIN From feet above a swimming pool, Susan throws a ball upward with a velocit of feet per second. The height h(t) of the ball t seconds after Susan throws it is given b h(t) = -1t + t +. For t 0, find the maimum height reached b the ball and the time that this height is reached. 0 ft; 1 s 11. HEALTH CLUBS Last ear, the SportsTime Athletic Club charged $0 to participate in an aerobics class. Sevent people attended the classes. The club wants to increase the class price this ear. The epect to lose one customer for each $1 increase in the price. a. What price should the club charge to maimize the income from the aerobics classes? $ b. What is the maimum income the SportsTime Athletic Club can epect to make? $0-1 Word Problem Practice Graphing Quadratic Functions 1. TRAJECTRIES A cannonball is launched from a cannon on the wall of Fort Chambl, Quebec. If the path of the cannonball is traced on a piece of graph paper aligned so that the cannon is situated on the -ais, the equation that describes the path is 1 = , where is the horizontal distance from the cliff and is the vertical distance above the ground in feet. How high above the ground is the cannon? 0 ft. TICKETING The manager of a smphon computes that the smphon will earn -0P P dollars per concert if the charge P dollars for tickets. What ticket price should the smphon charge in order to maimize its profits? $1.7. ARCHES An architect decides to use a parabolic arch for the main entrance of a science museum. In one of his plans, the top edge of the arch is described b the graph of = What are the coordinates of the verte of this parabola? (, ). FRAMING A frame compan offers a line of square frames. If the side length of the frame is s, then the area of the opening in the frame is given b the function a(s) = s - 10s +. Graph a(s). Chapter 9 Glencoe Algebra a. WALKING Canal Street and Walker Street are perpendicular to each other. Evita is driving south on Canal Street and is currentl miles north of the intersection with Walker Street. Jack is at the intersection of Canal and Walker Streets and heading east on Walker. Jack and Evita are both driving 0 miles per hour. a. When Jack is miles east of the intersection, where is Evita? - mi north of the intersection b. The distance between Jack and Evita is given b the formula + ( - ). For what value of are Jack and Evita at their closest? (Hint: Minimize the square of the distance.) =. c. What is the distance of closest approach? mi s Lesson -1 Answers (Lesson -1)

4 Chapter A Glencoe Algebra -1 Enrichment Finding the -intercepts of a Parabola As ou know, if f() = a + b + c is a quadratic function, the values of that make f() equal to zero are -b + b - ac and -b - b - ac. a a The average of these two number values is - b a. The function f() has its maimum or minimum value when = - b. The -intercepts of the parabola, a when the eist, are b - ac units to the left and a right of the ais of smmetr. Chapter 10 Glencoe Algebra f() b = a ( ( f() = a + b + c b, f b a a Eample Find the verte, ais of smmetr, and -intercepts for f() = Use = - b a. = - 10 () = -1 The -coordinate of the verte is -1. Substitute = -1 in f() = f(-1) = (-1) + 10(-1) - 7 = -1. The verte is (-1,-1). The ais of smmetr is = - b, or = -1. a The -coordinates of the -intercepts are 1 ± b - ac = 1 ± 10 - (-7) a = 1 ± The intercepts are ( 1 1, 0) and ( 1 + 1, 0). Eercises Find the verte, ais of smmetr, and -intercepts for the graph of each function using = - b a. 1. f() = - - (, -1); =. g() = (-1, 7); = -1. = (, 19); =. f() = + +. A() = (-, 0); = -. k() = (-, 1 ; = - ) ( 1, - 1 ; = ) 1 (( - Stud Guide and Intervention Solving Quadratic Equations b Graphing Solve Quadratic Equations Quadratic Equation A quadratic equation has the form a + b + c = 0, where a 0. Roots of a Quadratic Equation solution(s) of the equation, or the zero(s) of the related quadratic function The zeros of a quadratic function are the -intercepts of its graph. Therefore, finding the -intercepts is one wa of solving the related quadratic equation. Eample Solve + - = 0 b graphing. Graph the related function f() = + -. The -coordinate of the verte is -b a = - 1, and the equation of the ais of smmetr is = - 1. Make a table of values using -values around f() From the table and the graph, we can see that the zeros of the function are and -. Eercises Use the related graph of each equation to determine its solution = 0, = 0, = 0 1, f() f() f() = = = f() - -, 7 no real solutions - 1 Chapter 11 Glencoe Algebra f() f() f() Lesson - Answers (Lesson -1 and Lesson -)

5 Chapter A Glencoe Algebra - Stud Guide and Intervention (continued) Solving Quadratic Equations b Graphing Estimate Solutions ften, ou ma not be able to find eact solutions to quadratic equations b graphing. But ou can use the graph to estimate solutions. Eample Solve - - = 0 b graphing. If eact roots cannot be found, state the consecutive integers between which the roots are located. The equation of the ais of smmetr of the related function is = - - = 1, so the verte has -coordinate 1. Make a table of values. (1) f() The -intercepts of the graph are between and and between 0 and -1. So one solution is between and, and the other solution is between 0 and -1. Eercises Solve the equations. If eact roots cannot be found, state the consecutive integers between which the roots are located = = = 0 f() Chapter 1 Glencoe Algebra f() - f() - - f() - - between 0 and 1; between - and -1; between -1 and 0; between and between - and - between - and = = = 0 f() - - f() f() between and ; between and ; between - and -1; between - and -1 between and between and - Skills Practice Solving Quadratic Equations B Graphing Use the related graph of each equation to determine its solutions = = = 0 f() f() = + - f() = Chapter 1 Glencoe Algebra f() f() = + + -, 1 - no real solutions Solve each equation. If eact roots cannot be found, state the consecutive integers between which the roots are located = = = 0 f() f() = - + f() f() = = = 0 f() =- - f() f() 1 f() 1 1 f() 1 f() = - + 1, no real solutions between 0 and 1; between and 0, - -, f() =- + Lesson - Answers (Lesson -)

6 Chapter A Glencoe Algebra - Practice Solving Quadratic Equations B Graphing Use the related graph of each equation to determine its solutions = = = 0 f() f() f() , - - no such real numbers eist Chapter 1 Glencoe Algebra -1, 1 no real solutions 1, - - Solve each equation. If eact roots cannot be found, state the consecutive integers between which the roots are located = = = f() = = 0 f() f() f() f() between 0 and 1; -, - between - and -1, between - and - 9. GRAVITY Use the formula h(t) = v 0 t - 1t, where h(t) is the height of an object in feet, v 0 is the object s initial velocit in feet per second, and t is the time in seconds. a. Marta throws a baseball with an initial upward velocit of 0 feet per second. Ignoring Marta s height, how long after she releases the ball will it hit the ground?.7 s b. A volcanic eruption blasts a boulder upward with an initial velocit of 0 feet per second. How long will it take the boulder to hit the ground if it lands at the same elevation from which it was ejected? 1 s - Word Problem Practice Solving Quadratic Equations b Graphing 1. TRAJECTRIES David threw a baseball into the air. The function of the height of the baseball in feet is h = 0t -1t, where t represents the time in seconds after the ball was thrown. Use this graph of the function to determine how long it took for the ball to fall back to the ground. a seconds. BRIDGES In 19, a brick arch railwa bridge was built on North Avenue in Baltimore, Marland. The arch is described b the equation h = 9 1 0, where h is the height in ards and is the distance in ards from the center of the bridge. Graph this equation and describe, to the nearest ard, where the bridge touches the ground ards from the center of the bridge on either side at (-1, 0) and (1, 0). LGIC Wilma is thinking of two numbers. The sum is and the product is -. Use a quadratic equation to find the two numbers. and - t. RADI TELESCPES The cross-section of a large radio telescope is a parabola. The dish is set into the ground. The equation that describes the cross-section is d = 7 - -, where d gives the depth of the dish below ground and is the distance from the control center, both in meters. If the dish does not etend above the ground level, what is the diameter of the dish? Solve b graphing. m Chapter 1 Glencoe Algebra -0 0 f() BATS The distance between two boats is d = t - 10t +, where d is distance in meters and t is time in seconds. a. Make a graph of d versus t d t b. Do the boats ever collide? No Lesson - Answers (Lesson -)

7 Chapter A7 Glencoe Algebra - Enrichment Graphing Absolute Value Equations You can solve absolute value equations in much the same wa ou solved quadratic equations. Graph the related absolute value function for each equation using a graphing calculator. Then use the ZER feature in the CALC menu to find its real solutions, if an. Recall that solutions are points where the graph intersects the -ais. For each equation, make a sketch of the related graph and find the solutions rounded to the nearest hundredth = = = 0 - No solutions = = = 0-11, -9, -1, 7. + =. + 1 = = 0 -, - -, - -, 10. Eplain how solving absolute value equations algebraicall and finding zeros of absolute value functions graphicall are related. Sample answer: values of when solving algebraicall are the -intercepts (or zeros) of the function when graphed. Chapter 1 Glencoe Algebra - Stud Guide and Intervention Solving Quadratic Equations b Factoring Factored Form To write a quadratic equation with roots p and q, let ( - p)( - q) = 0. Then multipl using FIL. Eample a., - Write a quadratic equation in standard form with the given roots. ( - p)( - q) = 0 Write the pattern. ( - )[ - (-)] = 0 Replace p with, q with -. ( - )( + ) = 0 Simplif = 0 Use FIL. The equation = 0 has roots and -. Eercises b. - 7, 1 ( - p)( - q) = 0 [ - (- 7 ) ] ( - 1 = 0 ) ( + 7 ) ( - 1 = 0 ) ( + 7) ( - 1) = 0 ( + 7)( - 1) = = 0 The equation = 0 has roots - 7 and 1. Write a quadratic equation in standard form with the given root(s). 1., -. -, -. 1, = = = , 7. -, = = = ,., 9. -7, = = = 0 10., , , = = = 0 1., - 1., , = = = , , 1. 1, = = = 0 Chapter 17 Glencoe Algebra Lesson - Answers (Lesson - and Lesson -)

8 Chapter A Glencoe Algebra - Stud Guide and Intervention (continued) Solving Quadratic Equations b Factoring Solve Equations b Factoring When ou use factoring to solve a quadratic equation, ou use the following propert. Zero Product Propert For an real numbers a and b, if ab = 0, then either a = 0 or b =0, or both a and b = 0. Eample a. = 1 Solve each equation b factoring. = 1 riginal equation - 1 = 0 Subtract 1 from both sides. ( - ) = 0 Factor the binomial. = 0 or - = 0 Zero Product Propert = 0 or = Solve each equation. The solution set is {0, }. Eercises Solve each equation b factoring. b. - = 1 - = 1 riginal equation = 0 Subtract 1 from both sides. ( + 7)( - ) = 0 Factor the trinomial. + 7 = 0 or - = 0 Zero Product Propert = - 7 or = Solve each equation. The solution set is {- 7, } = 0. = 7. 0 = - {0, 1 } {0, 7} {0, - }. = = = 0 {0, 7 } {0, 9 } {0, } = = = 0 {, -} {, -1 } {-11, -} = = = 0 { 1, -7 } {-10, 1 } {- 1, } = = = 0 { 1, 1 } {, - } {100, } = = = 0 {, - } {-11, 1 } { 7, - } = = = 0 {, 1 } {-, 1 } {, - } = = = 0 7, } {-, - } {- 1, -1 } { 1 Chapter 1 Glencoe Algebra - Skills Practice Solving Quadratic Equations b Factoring Write a quadratic equation in standard form with the given root(s). 1. 1, - + = 0., = 0. -, = 0. 0, 7-7 = , = , - - = 0 Factor each polnomial. 7. m + 7m (m - )(m + 9) ( - )( + 1) 9. z + z p + p - (z + )(z - ) (p - )(p + ) c ( + )( + ) (c + 10)(c - 10) Solve each equation b factoring. 1. = {-, } = 0 {10, -10} = 0 {1, } = 0 {1, } = 0 {1, -} = 0 {, -} = 0 {1, } = 0 {0, 9} 1. - = 1 {-, 7}. + - = 0 { 1, - }. + - = 0 { = 0, - } {-, }. NUMBER THERY Find two consecutive integers whose product is 7. 1, 17 or -1, -17 Chapter 19 Glencoe Algebra Lesson - Answers (Lesson -)

9 Chapter A9 Glencoe Algebra - Practice Solving Quadratic Equations b Factoring Write a quadratic equation in standard form with the given root(s). 1. 7,. 0,. -, = 0 - = = 0. -7, -. -, -., = = = , 1. 1, 9. 0, = = = 0 Factor each polnomial. 10. r + r - r 11. a + a - 1. c - 9 r(r + 9)(r - ) (a - )(a + 1) (c - 7)(c + 7) r b - 1 ( + )( - + ) (r + 1)(r - 1) (b + 9)(b + )(b - ) Solve each equation b factoring = 0 {, -} = 0 {} = 0 {, } = 0 {-, -1} 0. - = 0 {0, } 1. 7 = {0,. 10 = 9. = + 99 {0, {-9, 11} 10} = - {-} = 0 {, }. = = 0 { {9, -}, - }. NUMBER THERY Find two consecutive even positive integers whose product is., 9. NUMBER THERY Find two consecutive odd positive integers whose product is. 17, GEMETRY The length of a rectangle is feet more than its width. Find the dimensions of the rectangle if its area is square feet. 7 ft b 9 ft 1. PHTGRAPHY The length and width of a -inch b -inch photograph are reduced b the same amount to make a new photograph whose area is half that of the original. B how man inches will the dimensions of the photograph have to be reduced? in. Chapter 0 Glencoe Algebra 7} - Word Problem Practice Solving Quadratic Equations b Factoring 1. FLASHLIGHTS When Dora shines her flashlight on the wall at a certain angle, the edge of the lit area is in the shape of a parabola. The equation of the parabola is = Factor this quadratic equation. ( - )( + ). SIGNS David was looking through an old algebra book and came across this equation. + = 0 The sign in front of the was blotted out. How does the missing sign depend on the signs of the roots? The missing sign is the same as the sign of the two roots, because their product is a positive number,.. ART The area in square inches of the drawing Maisons prés de la mer b Claude Monet is approimated b the equation = Factor the equation to find the two roots, which are equal to the approimate length and width of the drawing. 10 inches b 1 inches. PRGRAMMING Ra is a computer programmer. He needs to find the quadratic function of this graph for an algorithm related to a game involving dice. Provide such a function. Chapter 1 Glencoe Algebra f() = ANIMATIN A computer graphics animator would like to make a realistic simulation of a tossed ball. The animator wants the ball to follow the parabolic trajector represented b the quadratic equation f() = -0.( + ) ( - ). a. What are the solutions of f() = 0? = - or = b. Write f() in standard form. f() = c. If the animator changes the equation to f() = , what are the solutions of f() = 0? = -10 or = 10 Lesson - Answers (Lesson -)

10 Chapter A10 Glencoe Algebra - Enrichment Using Patterns to Factor Stud the patterns below for factoring the sum and the difference of cubes. a + b = (a + b)(a - ab + b ) a - b = (a - b)(a + ab + b ) This pattern can be etended to other odd powers. Stud these eamples. Eample 1 Factor a + b. Etend the first pattern to obtain a + b = (a + b)(a - a b + a b - ab + b ). Check: (a + b)(a - a b + a b - ab + b ) = a - a b + a b - a b + ab Eample Factor a - b. + a b - a b + a b - ab + b = a + b Etend the second pattern to obtain a - b = (a - b)(a + a b + a b + ab + b ). Check: (a - b) (a + a b + a b + ab + b ) = a + a b + a b + a b + ab - a b - a b - a b - ab - b = a - b In general, if n is an odd integer, when ou factor a n + b n or a n - b n, one factor will be either (a + b) or (a - b), depending on the sign of the original epression. The other factor will have the following properties: The first term will be a n - 1 and the last term will be b n - 1. The eponents of a will decrease b 1 as ou go from left to right. The eponents of b will increase b 1 as ou go from left to right. The degree of each term will be n - 1. If the original epression was a n + b n, the terms will alternatel have + and - signs. If the original epression was a n - b n, the terms will all have + signs. Use the patterns above to factor each epression. 1. a 7 + b 7 (a + b)(a - a b + a b - a b + a b - ab + b ). c 9 - d 9 (c - d)(c + c 7 d + c d + c d + c d + c d + c d + cd 7 + d ). f 11 + g 11 (f + g)(f 10 - f 9 g + f g - f 7 g + f g - f g + f g - f g 7 + f g - fg 9 + g 10 ) To factor 10-10, change it to ( + )( - ) and factor each binomial. Use this approach to factor each epression ( + )( )( - )( ). a 1 - b 1 (a + b)(a - a b + a b - a b + a b - ab + b ) (a - b) (a + a b + a b + a b + a b + ab + b ) Chapter Glencoe Algebra - Graphing Calculator Activit Using Tables to Factor b Grouping The TABLE feature of a graphing calculator can be used to help factor a polnomial of the form a + b + c. (The same problems can be solved with the Lists and Spreadsheet application on the TI-Nspire.) Eample 1 Factor b grouping. Make a table of the negative factors of 10 or 0. Look for a pair of factors whose sum is -. Enter the equation = 0 in Y1 to find the factors of 0. Then, find the sum of the factors using = 0 + in Y. Set up the table to displa the negative factors of 0 b setting Tbl = to -1. Eamine the results. Kestrokes: Y= 0 ENTER VARS ENTER ENTER + ENTER nd [TBLSET] ( ) 1 ENTER ( ) 1 ENTER nd [TABLE]. The last line of the table shows that - ma be replaced with - + (-) = (-) + = ( - ) + (-7)( - ) = ( - )( - 7) Thus, = ( - )( - 7). Eample Factor each polnomial. Factor Look at the factors of 1(-1) or -1 for a pair with a sum of -7. Enter an equation to determine the factors in Y1 and an equation to find the sum of factors in Y. Eamine the table to find a sum of -7. Kestrokes: Y= ( ) 1 ENTER VARS ENTER ENTER + ENTER nd [TBLSET] 1 ENTER 1 ENTER nd [TABLE] = (-1) - 1 = ( + ) - ( + ) = ( + )( - ) Thus, = ( + )( - ). Eercises z - z a + a - 1 ( + )( - ) (z - )(z - 7) ( + 9)( - ) prime. m + 17m + 1. z - z b + b - 0 (m + )(m + ) (1z - )(z - ) ( + 7) (b + 1)(b - 1) Chapter Glencoe Algebra Lesson - Answers (Lesson -)

11 Chapter A11 Glencoe Algebra - Stud Guide and Intervention Comple Numbers Pure Imaginar Numbers A square root of a number n is a number whose square is n. For nonnegative real numbers a and b, ab = a b and a b = a, b 0. b The imaginar unit i is defined to have the propert that i = -1. Simplified square root epressions do not have radicals in the denominator, and an number remaining under the square root has no perfect square factor other than 1. Eample 1 Eample a. Simplif -. - = 1 (-) = 1-1 = i b. Simplif -. - = = = i 7 Eample Solve + = 0. + = 0 riginal equation. = - Subtract from each side. = ± i Square Root Propert. Eercises Simplif i. - i. - i 1. ( + i) ( - i) Solve each equation. a. Simplif -i i. -i i = -1i = -1(-1) = 1 b. Simplif = i i 1 = i = -1 9 = -. + = 0 ±i. + = 0 ±i = 9 ± i. 7 + = 0 ±i Chapter Glencoe Algebra - Stud Guide and Intervention (continued) Comple Numbers perations with Comple Numbers Comple Number Addition and Subtraction of Comple Numbers Multiplication of Comple Numbers Comple Conjugate A comple number is an number that can be written in the form a + bi, where a and b are real numbers and i is the imaginar unit (i = -1). a is called the real part, and b is called the imaginar part. Combine like terms. (a + bi) + (c + di) = (a + c) + (b + d )i (a + bi) - (c + di) = (a - c) + (b - d )i Use the defi nition of i and the FIL method: (a + bi)(c + di) = (ac - bd ) + (ad + bc)i a + bi and a - bi are comple conjugates. The product of comple conjugates is alwas a real number. To divide b a comple number, first multipl the dividend and divisor b the comple conjugate of the divisor. Eample 1 Simplif ( + i) + ( - i). Eample ( + i) + ( - i) = ( + ) + (1 - )i = 10 - i Eample Simplif ( - i) (- + i). Eample ( - i) (- + i) = (-) + (i) + (-i)(-) + (-i)(i) = - + i + 0i - 10i = - + i - 10(-1) = + i Eercises Simplif. 1. (- + i) + ( - i). ( - i) - ( - i). ( - i) + ( - i) - i + i 10 - i. (-11 + i) - (1 - i). ( + i) + ( - i). ( + i) - (- - i) i i 7. ( + i)( - i) 7 + i. ( - i)( - i) 1-1i 9. ( - i)(1 - i) -10i i - 1 i i i ( + i) - ( - i) = ( - ) + [ - (-)]i = + i - i + i = - i + i Simplif ( + i) - ( - i). Simplif - i + i. - i - i i - 7 i i - - i Chapter Glencoe Algebra = - 9i - i + i + 9 i = - 11i 1 = i Lesson - Answers (Lesson -)

12 Chapter A1 Glencoe Algebra - Skills Practice Simplif. Comple Numbers i (i)(-i)(i) 0i. i 11 -i 9. i i 10. (7 - i) + (-1 - i) - - 1i 11. (- + i) + (1-7i) 1 - i 1. (10 - i) - (7 + i) - 7i 1. (7 - i)( - i) - - i 1. ( + i)( - i) 1. - i - - i i Solve each equation. 1. i + i = 0 ±i = 0 ±i = 0 ±i = 0 ±i = 0 ±i. + 9 = 0 ±i Find the values of l and m that make each equation true. i + i 10 1, i = l + (m)i, -. l - 1i = - (m)i,. ( + l) + (m)i = 9 + 1i, 7. ( - m) + (7l - 1)i = 1 + 7i, Chapter Glencoe Algebra - Practice Simplif. Comple Numbers i (-i) (i)(-i). (7i) (i). i -0i -9i i. i 9 9. ( - i) + (-1 - i) -i i i 10. (7 - i) + (9 + 11i) 11. (-1 + i) + (1 + 1i) 1. (10 + 1i) - ( - 0i) 1 + i + 9i - + i 1. ( - i) - (10-0i) 1. ( - i) ( + i) 1. ( - 11i) ( - 11i) 1 + i -7-17i 1. ( + i) ( - i) 17. (7 + i) (9 - i) 1. + i - + i -i - 1i 7 - i i 1 + 1i 11 Solve each equation i - i. n + = 0 ±i 7. m + 10 = i 1 + i. m + 7 = 0 ±i 19. -m - = 0 ±i ±i 7 + i. -m - = = 0 ±i Find the values of l and m that make each equation true. ±i -1 - i. 1 - i = l + (m)i, ( - l) + (m)i = i 1, 9 0. (l + ) + ( - m)i = 1 - i, 1. (7 + m) + (l - 10)i = - i 1, -. ELECTRICITY The impedance in one part of a series circuit is 1 + j ohms and the impedance in another part of the circuit is 7 - j ohms. Add these comple numbers to find the total impedance in the circuit. - j ohms. ELECTRICITY Using the formula E = IZ, find the voltage E in a circuit when the current I is - j amps and the impedance Z is + j ohms j volts Chapter 7 Glencoe Algebra Lesson - Answers (Lesson -)

13 Chapter A1 Glencoe Algebra - Word Problem Practice Comple Numbers 1. SIGN ERRRS Jennifer and Jessica come up with different answers to the same problem. The had to multipl ( + i)( - i) and give their answer as a comple number. Jennifer claims that the answer is 1 and Jessica claims that the answer is 17. Who is correct? Eplain. Jessica is correct; ( + i)( - i) = 1 + i - i - i = 1 - (-1) = = 17.. CMPLEX CNJUGATES You have seen that the product of comple conjugates is alwas a real number. Show that the sum of comple conjugates is also alwas a real number. a + bi and a - bi are comple conjugates and their sum is a, which is real.. PYTHAGREAN TRIPLES If three integers a, b, and c satisf a + b = c, then the are called a Pthagorean triple. Suppose that a, b, and c are a Pthagorean triple. Show that the real and imaginar parts of (a + bi), together with the number c, form another Pthagorean triple. (a + bi) = a - b + abi; a - b and ab are integers and (a - b ) + (ab) = a - a b + b + a b = a + a b + b = (a + b ) = (c ), so a + b = c as desired.. RTATINS Comple numbers can be used to perform rotations in the plane. For eample, if (, ) are the coordinates of a point in the plane, then the real and imaginar parts of i( + i) are the horizontal and vertical coordinates of the 90 counterclockwise rotation of (, ) about the origin. What are the real and imaginar parts of i( + i)? The real part is - and imaginar part is.. ELECTRICAL ENGINEERING Alternating current (AC) in an electrical circuit can be described b comple numbers. In an electrical circuit, Z, the impedance in the circuit, is related to the voltage V and the current I b the formula Z = V. The standard electrical I voltage in Europe is 0 volts, so in these problems use V = 0. a. Find the impedance in a standard European circuit if the current is 11i amps. + i b. Find the current in a standard European circuit if the impedance is 10 i watts. 1 i amps c. Find the impedance in a standard European circuit if the current is 0i amps. 11i amps Chapter Glencoe Algebra - Enrichment Conjugates and Absolute Value When studing comple numbers, it is often convenient to represent a comple number b a single variable. For eample, we might let z = + i. We denote the conjugate of z b z. Thus, z = - i. We can define the absolute value of a comple number as follows. z = + i = + There are man important relationships involving conjugates and absolute values of comple numbers. Eample 1 Let z = + i. Then, zz z = ( + i)( - i) = + = ( + ) = z Eample comple number z. Show z = z z for an comple number z. Show z is the multiplicative inverse for an nonzero z We know z = z z. If z 0, then we have z ( z z ) = 1. Thus, z is the multiplicative inverse of z. z Eercises For each of the following comple numbers, find the absolute value and multiplicative inverse. 1. i ; -i - + i 1 + i. - - i ;. 1 - i 1; i 1; + 1i. 1 + i ; 1 - i. - i ; + i i. - i i ; - i 1; + i 1; 1 + i Chapter 9 Glencoe Algebra Lesson - Answers (Lesson -)

14 Chapter A1 Glencoe Algebra - Stud Guide and Intervention Completing the Square Square Root Propert Use the Square Root Propert to solve a quadratic equation that is in the form perfect square trinomial = constant. Eample Solve each equation b using the Square Root Propert. Round to the nearest hundredth if necessar. a = = ( - ) = - = or - = - = + = 9 or = - + = -1 The solution set is {9, -1}. Eercises b = = ( - ) = - = or - = - - = or - = - = ± The solution set is { ± }. Solve each equation b using the Square Root Propert. Round to the nearest hundredth if necessar = = = 1 {, 1} {-, -1} {, = = = { -1 ± - ± 7 } } { 0, } { = = = 11 { 1, - 1, - } -{0., 1.} } { = = = 9 { - ± } { - ± } Chapter 0 Glencoe Algebra } } { ± - Stud Guide and Intervention (continued) Completing the Square Complete the Square To complete the square for a quadratic epression of the form + b, follow these steps. 1. Find b.. Square b.. Add ( b ) to + b. Eample 1 Find the value Eample Solve - - = 0 b of c that makes + + c a perfect square trinomial. Then write the trinomial as the completing the square. - - = 0 riginal equation square of a binomial. - - = 0 Divide each side b. Step 1 b = ; b = 11 Step 11 = 11 Step c = 11 The trinomial is , which can be written as ( + 11). Eercises = 0 - = is not a perfect square. Add 1 to each side. - + = 1 + Since ( ) =, add to each side. ( - ) = 1 Factor the square. - = ± Square Root Propert = or = - Solve each equation. The solution set is {, -}. Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square c c. - + c ; ( - ) 900; ( + 0) 9 ; ( c c c.; ( + 1.) 1 1 ; ( + 1 ) 1.; ( - 1.) Solve each equation b completing the square = = 0 9. w - 10w + 1 = 0-1, -, 1, = = = 0 1, 1-1, 7 1, = = 0 1. t + t - = 0 - ± - ± - ± 1 Chapter 1 Glencoe Algebra ) Lesson - Answers (Lesson -)

15 Chapter A1 Glencoe Algebra - Skills Practice Completing the Square Solve each equation b using the Square Root Propert. Round to the nearest hundredth if necessar = 1,. + + = 1-1, = -1, = 9-1,. + + = -.1, = -1., = 7 0., = , -.1 Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square c ; ( + ) c 9; ( - 7) c 1; ( + 1) c ) c c ; ( - 9 Solve each equation b completing the square = 0, = 0 0, = 0, = = 0 -, = 0 - ± = = 0. = -11 ± i = 0 ; ( + ) 1 ; ( - 1 ) ± 17 1, -1 1 ± 1 1 ± i Chapter Glencoe Algebra - Practice Completing the Square Solve each equation b using the Square Root Propert. Round to the nearest hundredth if necessar = = = 1 -, - -, - -9, = = =, 10-1, - ± = = = ± 1 ± Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square c c c ; ( + ) 100; ( - 10) 11 ; ( + 11 ) c c c 0.1; ( + 0.) 1.1; ( - 1.1) 0.0; ( - 0.1) c c c 1 ; ( + 1) 1 ; ( - 1 ) ; ( - Solve each equation b completing the square = 0 -, = 0, = = = = 0, 7 ± 0 - ± = = = 0 - ± -1 ± 1 ± = = = 0 - ± i 1 - ± i - ± i 7 1. GEMETRY When the dimensions of a cube are reduced b inches on each side, the surface area of the new cube is square inches. What were the dimensions of the original cube? 1 in. b 1 in. b 1 in.. INVESTMENTS The amount of mone A in an account in which P dollars are invested for ears is given b the formula A = P(1 + r), where r is the interest rate compounded annuall. If an investment of $00 in the account grows to $ in two ears, at what interest rate was it invested? % Chapter Glencoe Algebra ) 1, Lesson - Answers (Lesson -)

16 Chapter A1 Glencoe Algebra - Word Problem Practice Completing the Square 1. CMPLETING THE SQUARE Samantha needs to solve the equation - 1 = 0. What must she do to each side of the equation to complete the square? Add.. ART The area in square inches of the drawing Foliage b Paul Cézanne is approimated b the equation = Complete the square and find the two roots, which are equal to the approimate length and width of the drawing. 1 inches b inches. CMPUND INTEREST Nikki invested $1000 in a savings account with interest compounded annuall. After two ears the balance in the account is $110. Use the compound interest formula A = P(1 + r) t to find the annual interest rate. 10%. REACTIN TIME Lauren was eating lunch when she saw her friend Jason approach. The room was crowded and Jason had to lift his tra to avoid obstacles. Suddenl, a glass on Jason s lunch tra tipped and fell off the tra. Lauren lunged forward and managed to catch the glass just before it hit the ground. The height h, in feet, of the glass t seconds after it was dropped is given b h = -1t +.. Lauren caught the glass when it was si inches off the ground. How long was the glass in the air before Lauren caught it? 0. second. PARABLAS A parabola is modeled b = Jane s homework problem requires that she find the verte of the parabola. She uses the completing square method to epress the function in the form = ( - h) + k, where (h, k) is the verte of the parabola. Write the function in the form used b Jane. = ( - ) + = 0. AUDITRIUM SEATING The seats in an auditorium are arranged in a square grid pattern. There are rows and columns of chairs. For a special concert, organizers decide to increase seating b adding n rows and n columns to make a square pattern of seating + n seats on a side. a. How man seats are there after the epansion? n + 90n + 0 b. What is n if organizers wish to add 1000 seats? 10 c. If organizers do add 1000 seats, what is the seating capacit of the auditorium? 0 Chapter Glencoe Algebra - Enrichment The Golden Quadratic Equations A golden rectangle has the propert that its length can be written as a + b, where a is the width of the rectangle and a + a b = a. An golden rectangle can be b divided into a square and a smaller golden rectangle, as shown. The proportion used to define golden rectangles can be used to derive two quadratic equations. These are sometimes called golden quadratic equations. Solve each problem. 1. In the proportion for the golden rectangle, let a equal 1. Write the resulting quadratic equation and solve for b. b + b - 1 = 0 b = In the proportion, let b equal 1. Write the resulting quadratic equation and solve for a. a - a - 1 = 0 a = 1 +. Describe the difference between the two golden quadratic equations ou found in eercises 1 and. The signs of the first-degree terms are opposite.. Show that the positive solutions of the two equations in eercises 1 and are reciprocals. ( -1 + ) = -( 1 ) + ( ) = -1 + = 1 ) ( 1 +. Use the Pthagorean Theorem to find a radical epression for the diagonal of a golden rectangle when a = 1. d = Find a radical epression for the diagonal of a golden rectangle when b = 1. d = 10 + Chapter Glencoe Algebra a a a b b a Lesson - Answers (Lesson -)

17 Chapter A17 Glencoe Algebra - Stud Guide and Intervention The Quadratic Formula and the Discriminant Quadratic Formula The Quadratic Formula can be used to solve an quadratic equation once it is written in the form a + b + c = 0. Quadratic Formula Eample The solutions of a + b + c = 0, with a 0, are given b = -b ± b - ac. a Solve - = 1 b using the Quadratic Formula. Rewrite the equation as = 0. = -b ± b - ac Quadratic Formula a = -(-) ± (-) - (1)(-1) Replace a with 1, b with -, and c with -1. (1) = ± 1 Simplif. = ± 9 = 7 or - The solutions are - and 7. Eercises Solve each equation b using the Quadratic Formula = = = 0, -7 -, -, = = = 0 1, - - 1, - 1, = = = 0 -, 1, -, = r - r + = = 0 -,, 1 ± = = = 0 - ± ± ± i Chapter Glencoe Algebra - Roots and the Discriminant Discriminant Discriminant b - ac > 0 and a perfect square b - ac > 0, but not a perfect square b - ac = 0 b - ac < 0 The epression under the radical sign, b - ac, in the Quadratic Formula is called the discriminant. Tpe and Number of Roots rational roots irrational roots 1 rational root comple roots Eample Find the value of the discriminant for each equation. Then describe the number and tpe of roots for the equation. a. + + The discriminant is b - ac = - () () or 1. The discriminant is a perfect square, so the equation has rational roots. Eercises Stud Guide and Intervention b. - + The discriminant is b - ac = (-) - () () or -. The discriminant is negative, so the equation has comple roots. Complete parts a-c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and tpe of roots. c. Find the eact solutions b using the Quadratic Formula. (continued) The Quadratic Formula and the Discriminant 1. p + 1p = = = 0 irrational roots; 1 rational root; 1 - ± rational roots; - 1,. + - = = = 0 irrational roots; rational roots; -10; comple roots; - ± 1, 7 1 ± i = 0. m - m = = -1 rational roots; irrational roots; 1 rational root; 1, ± = = = 0 comple roots; rational roots; irrational roots; - ± i -, ± Chapter 7 Glencoe Algebra Lesson - Answers (Lesson -)

18 Chapter A1 Glencoe Algebra - Skills Practice The Quadratic Formula and the Discriminant Complete parts a-c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and tpe of roots. c. Find the eact solutions b using the Quadratic Formula = = 0 0; 1 rational root; ; rational roots; -, 1. - = = 0 ; rational roots; 0, 9; rational roots; -, 1. - = 0. - = 0 10; irrational roots; ± ; irrational roots; ± = = 0 1; irrational roots; - ± 1; irrational roots; 1 ± = = 0 7; irrational roots; 1 ± -19; comple roots; ±7i = = - -; comple roots; 1 ± i -7; comple roots; ± i 7 Solve each equation b using the Quadratic Formula. 1. = ± = 0 ± = 0 -, = 0, = 0 ± = 0 ± = 0 ±i 0. + = 0 ± i = 0 - ± 7 ± = = 1 ± i. + + = 0-1 ± i. PARACHUTING Ignoring wind resistance, the distance d(t) in feet that a parachutist falls in t seconds can be estimated using the formula d(t) = 1t. If a parachutist jumps from an airplane and falls for 1100 feet before opening her parachute, how man seconds pass before she opens the parachute? about. s Chapter Glencoe Algebra - Practice The Quadratic Formula and the Discriminant Solve each equation b using the Quadratic Formula = 0 0,. - 9 = 0 ± 7. + = =, - -, = 0,. 1 + = - -, = 0 ± = 0 7 ± i ± 9. = - ± i = 0 1 ± i ± = = ± 1. = - 1 ± i = 0 Complete parts a-c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and tpe of roots. c. Find the eact solutions b using the Quadratic Formula = 0 1. = = 0 0; 1 rational; 9; rational; 0, 0; 1 rational; 1. - = = = 0 19; rational; -, irrational; -9 ± 10 9; rational; 0, = = = 0 comple; 1 ± i 19 rational;, - - ± i comple; = = = 0 rational; 1, - irrational; 1 ± 7 - ± i comple; = = = 0 irrational; ± 10 0; 1 rational; 1 irrational; ± 7 0. GRAVITATIN The height h(t) in feet of an object t seconds after it is propelled straight up from the ground with an initial velocit of 0 feet per second is modeled b the equation h(t) = -1t + 0t. At what times will the object be at a height of feet? 1.7 s, s 1. STPPING DISTANCE The formula d = 0.0s + 1.1s estimates the minimum stopping distance d in feet for a car traveling s miles per hour. If a car stops in 00 feet, what is the fastest it could have been traveling when the driver applied the brakes? about. mi/h Chapter 9 Glencoe Algebra Lesson - Answers (Lesson -)

19 Chapter A19 Glencoe Algebra - Word Problem Practice The Quadratic Formula and the Discriminant 1. PARABLAS The graph of a quadratic equation of the form = a + b + c is shown below. - Is the discriminant b - ac positive, negative, or zero? negative. TANGENT Kathleen is tring to find b so that the -ais is tangent to the parabola = + b +. She finds one value that works, b =. Is this the onl value that works? Eplain. No, b = - also works; the -ais will be tangent when the discriminant b - 1 is zero. This happens when b = or -.. SPRTS In 1990, American Rand Barnes set the world record for the shot put. His throw can be described b the equation = 1 +. Use the Quadratic Formula to find how far his throw was to the nearest foot. ft. EXAMPLES Give an eample of a quadratic function f() that has the following properties. I. The discriminant of f is zero. II. There is no real solution of the equation f() = 10. Sketch the graph of = f(). Sample answer: f() = Chapter 0 Glencoe Algebra TANGENTS The graph of = is a parabola that passes through the point at (1, 1). The line = m - m + 1, where m is a constant, also passes through the point at (1, 1). a. To find the points of intersection between the line = m - m + 1 and the parabola =, set = m - m + 1 and then solve for. Rearranging terms, this equation becomes - m + m - 1 = 0. What is the discriminant of this equation? - m + b. For what value of m is there onl one point of intersection? Eplain the meaning of this in terms of the corresponding line and the parabola. m = ; the parabola = and the line = - 1 have eactl one point of intersection at (1, 1). In other words, this line is tangent to the parabola at (1, 1). - Enrichment Sum and Product of Roots Sometimes ou ma know the roots of a quadratic equation without knowing the equation itself. Using our knowledge of factoring to solve an equation, ou can work backward to find the quadratic equation. The rule for finding the sum and product of roots is as follows: Sum and Product of Roots Eample The roots are = and = -. + (-) = - Add the roots. (-) = - Multipl the roots. Equation: + - = 0 Eercises If the roots of a + b + c = 0, with a 0, are s 1 and s, then s 1 + s = - a b and s s = c 1 a. Write a quadratic equation that has the roots and -. Write a quadratic equation that has the given roots. Chapter 1 Glencoe Algebra (, 0 1 ) 1., -9., -1., + - = = = 0. ±. -,. - ± = = = 0 Find k such that the number given is a root of the equation. 7. 7; + k - 1 = 0. -; k = Lesson - Answers (Lesson -)

20 Chapter A0 Glencoe Algebra - Spreadsheet Activit Approimating the Real Zeros of Polnomials You have learned the Location Principle, which can be used to approimate the real zeros of a polnomial. 1 A B C f() D E Sheet 1 Sheet Sheet F G 0 H Chapter Glencoe Algebra I In the spreadsheet above, the positive real zero of ƒ() = - can be approimated in the following wa. Set the spreadsheet preference to manual calculation. The values in A and B are the endpoints of a range of values. The values in D through J are values equall in the interval from A to B. The formulas for these values are A, A + (B - A)/, A + *(B - A)/, A + *(B - A)/, A + *(B - A)/, A + *(B - A)/, and B, respectivel. Row gives the function values at these points. The function ƒ() = - is entered into the spreadsheet in Cell D as D^ -. This function is then copied to the remaining cells in the row. You can use this spreadsheet to stud the function values at the points in cells D through J. The value in cell F is positive and the value in cell G is negative, so there must be a zero between -1.7 and 0. Enter these values in cells A and B, respectivel, and recalculate the spreadsheet. (You will have to recalculate a number of times.) The result is a new table from which ou can see that there is a zero between 1.11 and Because these values agree to three decimal places, the zero is about 1.1. This can be verified b using algebra. B solving - = 0, we obtain = ±. The positive root is = ± = , which verifies the result. Eercises 1. Use a spreadsheet like the one above to approimate the zero of ƒ() = - to three decimal places. Then verif our answer b using algebra to find the eact value of the root. The spreadsheet gives = 0.7. B solving for algebraicall, =. So, the approimation is correct.. Use a spreadsheet like the one above to approimate the real zeros of f() = Round our answer to four decimal places. Then, verif our answer b using the quadratic formula. The process gives and to the nearest ten-thousandth. The quadratic formula gives = -1 ± and Use a spreadsheet like the one above to approimate the real zero of ƒ() = between -0. and to the nearest ten-thousandth J -7 Stud Guide and Intervention Transformations with Quadratic Functions Write Quadratic Equations in Verte Form A quadratic function is easier to graph when it is in verte form. You can write a quadratic function of the form = a + b + c in verte from b completing the square. Eample = = ( - ) + = ( - + 9) = ( - ) + 7 Write = in verte form. Then graph the function. The verte form of the equation is = ( - ) + 7. Eercises Write each equation in verte form. Then graph the function. Chapter Glencoe Algebra 1. = = +. = - + = ( - ) + 7 = ( + ) - 9 = ( - ) = = = = -( - ) + = ( - ) - 7 = (- 1) Lesson -7 Answers (Lesson - and Lesson -7)