Answers (Anticipation Guide and Lesson 9-1)

Answers (Anticipation Guide and Lesson 9-) Chapter Resources NAME DATE PERID 9 Anticipation Guide Quadratic and Eponential Functions Step Before ou begin Chapter 9 Read each statement. 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 A, D, o NS Statement. The graph of a quadratic function is a parabola.. The graph of + 7 will be a parabola opening downward since the coefficient of is positive.. A quadratic function s ais of smmetr is either the -ais or the -ais.. The graph of a quadratic function opening upward has no maimum value. 5. The -intercepts of the graph of a quadratic function are the solutions to the related quadratic equation. 6. All quadratic equations have two real solutions. 7. An quadratic epression can be written as a perfect square b a method called completing the square. 8. The quadratic formula can onl be used to solve quadratic equations that cannot be solved b factoring or graphing. 9. A function containing powers is called an eponential function. 0. Receiving compound interest on a bank account is one eample of eponential growth. STEP A o D A D D A A D A D D A Step After ou complete Chapter 9 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? For those statements that ou mark with a D, use a piece of paper to write an eample of wh ou disagree. Chapter 9 Glencoe Algebra Answers NAME DATE PERID Lesson 9-9- Stud Guide and Intervention Graphing Quadratic Functions Characteristics of Quadratic Functions Quadratic Function a function described b an equation of the form f() = a + b + c, where a 0 Eample: = + + 8 The parent graph of the famil of quadratic fuctions is =. Graphs of quadratic functions have a general shape called a parabola. A parabola opens upward and has a minimum point when the value of a is positive, and a parabola opens downward and has a maimum point when the value of a is negative. Eample Eample a. Use a table of values to graph = - +. - 6 0 - - - Graph the ordered pairs in the table and connect them with a smooth curve. b. What is the domain and range of this function? The domain (the -values) is all real numbers. The range (the -values) is all real numbers greater than or equal to -, which is the minimum. a. Use a table of values to graph = - - 6-7. -6-7 -5 - - - - - - 0-7 Graph the ordered pairs in the table and connect them with a smooth curve. b. What is the domain and range of this function? The domain (the -values is all real numbers. The range (the -values) is all real numbers less than or equal to, which is the maimum. Eercises Use a table of values to graph each function. Determine the domain and range.. = +. = - -. = - + D: { is a real D: { is a real D: { is a real number.} R: { } number.} R: { -} number.} R: { - } Chapter 9 5 Glencoe Algebra Chapter 9 A Glencoe Algebra

Answers (Lesson 9-) Lesson 9- NAME DATE PERID 9- Stud Guide and Intervention (continued) Graphing Quadratic Functions Smmetr and Vertices Parabolas have a geometric propert called smmetr. That is, if the figure is folded in half, each half will match the other half eactl. The vertical line containing the fold line is called the ais of smmetr. The ais of smmetr contains the minimum or maimum point of the parabola, the verte. Ais of Smmetr For the parabola = a + b + c, where a 0, Eample: The ais of smmetr of a is the ais of smmetr. the line = - b = + + 5 is the line = -. Eample a. Write the equation of the ais of smmetr. In = + +, a = and b =. Substitute these values into the equation of the ais of smmetr. = - b a = - () = - The ais of smmetr is = -. c. Identif the verte as a maimum or a minimum. Since the coefficient of the -term is positive, the parabola opens upward, and the verte is a minimum point. d. Graph the function. Consider the graph of = + +. b. Find the coordinates of the verte. Since the equation of the ais of smmetr is = - and the verte lies on the ais, the -coordinate of the verte is -. = + + riginal equation = (-) + (-) + Substitute. = () - + Simplif. = - The verte is at (-, -). =- (-,-) Eercises Consider each equation. Determine whether the function has maimum or minimum value. State the maimum or minimum value. What are the domain and range of the function? Find the equation of the ais of smmetr. Graph the function.. = +. = - - -. = + + min; (0, ); D: { all reals}, R: { }; = 0 ma; (-, 0); D: { all reals}, R: { 0}; = - min; (-, ); D: { all reals}, R: { }; = - Chapter 9 6 Glencoe Algebra NAME DATE PERID 9- Skills Practice Graphing Quadratic Functions Use a table of values to graph each function. State the domain the range.. = -. = - +. = - - 6 D = all reals R = { } D = all reals R = { } D = all reals R = { 7} Find the verte, the equation of the ais of smmetr, and the -intercept.. = - 8 + 6 5. = + + 6 6. = - - + (, -) ; = ; (0, 6) (-, ) ; = -; (0, 6) (-, 5) ; = -; (0, ) Consider each equation. a. Determine whether the function has maimum or minimum value. b. State the maimum or minimum value. c. What are the domain and range of the function? 7. = 8. = - - 5 9. = - + - minimum; (0, 0) ; D = all reals, R = { 0} minimum; (, -6) ; D = all reals, R = { -6} maimum; (, ) ; D = all reals, R = { } Graph each function. 0. f() = - - +. f() = + -. f() = - - + 6 f() f() f() Chapter 9 7 Glencoe Algebra Chapter 9 A Glencoe Algebra

Answers (Lesson 9-) Lesson 9- NAME DATE PERID 9- Practice Graphing Quadratic Functions Use a table of values to graph each function. Determine the domain and range.. = - +. = - 6 +. = - - 8-5 D: { all reals} D: { all reals} D: { all reals} R: { } R: { -6} R: { } Find the verte, the equation of the ais of smmetr, and the -intercept.. = - 9 5. = - + 8-5 6. - + (0, -9); = 0; (0, -9) (, ); = ; (0, -5) (0.5, 0); = 0.5; (0, ) Consider each equation. Determine whether the function has maimum or minimum value. State the maimum or minimum value. What are the domain and range of the function? 7. = 5 - + 8. = - + 5-0 9. = + - 9 min; (0.,.8); D: { all reals}, R: {.8} ma; (.5, -.75); D: { all reals}, R: { -.75} min; (-, - ); D: { all reals}, Graph each function. R: { - } 0. f() = - +. f() = - + 8 -. f() = + 8 + f() f() f(). BASEBALL A plaer hits a baseball into the outfield. The equation h = -0.005 + + gives the path of the ball, where h is the height and is the horizontal distance the ball travels. a. What is the equation of the ais of smmetr? = 00 b. What is the maimum height reached b the baseball? 5 ft c. An outfielder catches the ball three feet above the ground. How far has the ball traveled horizontall when the outfielder catches it? 00 ft Chapter 9 8 Glencoe Algebra Answers NAME DATE PERID 9- Word Problem Practice Graphing Quadratic Functions. LYMPICS lmpics were held in 896 and have been held ever four ears (ecept 96, 90, and 9). The winning height in men s pole vault at an number lmpiad can be approimated b the equation = 0.7 +. + 6. Complete the table to estimate the pole vault heights in each of the lmpic Games. Round our answers to the nearest tenth. Year lmpiad () 896 900 9 7 96 0 96 5 008 6 Height ( inches) 0.7 6. 7. 06.0 7.8 87.9. SFTBALL lmpic softball gold medalist Michele Smith pitches a curveball with a speed of 6 feet per second. If she throws the ball straight upward at this speed, the ball s height h (in feet) after t seconds is given b h = -6t + 6t. Find the coordinates of the verte of the graph of the ball s height and interpret its meaning. (, 6); After seconds, the ball reaches its highest point, 6 ft above the ground. 5. GEMETRY Tedd is building the rectangular deck shown below. + 6 - Source: National Securit Agenc. PHYSICS Mrs. Capwell s phsics class investigates what happens when a ball is given an initial push, rolls up, and then back down an inclined plane. The class finds that = - + 6 accuratel predicts the ball s position after rolling seconds. n the graph of the equation, what would be the value when =? 8 a. Write the equation representing the area of the deck. = ( - )( + 6) or = + - b. What is the equation of the ais of smmetr? = - c. Graph the equation and label its verte.. ARCHITECTURE A hotel s main entrance is in the shape of a parabolic arch. The equation = - + 0 models the arch height for an distance from one side of the arch. Use a graph to determine its maimum height. 5 ft - -6-8 -0 - - -6-5--- (-,-6) Chapter 9 9 Glencoe Algebra Chapter 9 A Glencoe Algebra

Answers (Lesson 9- and Lesson 9-) Lesson 9- NAME DATE PERID 9- Enrichment Graphing Cubic Functions A cubic function is a polnomial written in the form of f() = a + b + c + n, where a 0. Cubic functions do not have absolute minimum and maimum values like quadratic functions do, but the can have a local minimum and a local maimum point. f() Parent Function: f() = Domain: all real numbers Range: all real numbers Eample Use a table of values to graph = + -. Then use the graph to estimate the locations of the local minimum and local maimum points. 0 (-, ) Graph the ordered pairs, and connect them to create a smooth curve. The S shape etends to infinit in the positive direction and to negative infinit in the negative direction. The local minimum is located at (0, ). The local maimum is located at (, ). (0, -) Eercises Use a table of values to graph each equation. Then use the graph to estimate the locations of the local minimum and local maimum points.. = 0.5 + -. = - - -. = + + - local maimum: local maimum: local maimum: (-., -0.); (0, -); (-.8, -.9); local minimum: local minimum: local minimum: (0, ) (-, -) (-0., -.) Chapter 9 0 Glencoe Algebra NAME DATE PERID 9- Stud Guide and Intervention Solving Quadratic Equations b Graphing Solve b Graphing Quadratic Equation an equation of the form a + b + c = 0, where a 0 The solutions of a quadratic equation are called the roots of the equation. The roots of a quadratic equation can be found b graphing the related quadratic function f() = a + b + c and finding the -intercepts or zeros of the function. Eample Solve + + = 0 b Eample graphing. Graph the related function f() = + +. The equation of the ais of smmetr is = - () or -. The verte is at (-, -). Graph the verte and several other points on either side of the ais of smmetr. Solve - 6 + 9 = 0 b graphing. Graph the related function f() = - 6 + 9. The equation of the ais of smmetr is = 6 () or. The verte is at (, 0). Graph the verte and several other points on either side of the ais of smmetr. f() f() To solve + + = 0, ou need to know where the value of f() = 0. This occurs at the -intercepts, - and -. The solutions are - and -. To solve - 6 + 9 = 0, ou need to know where the value of f() = 0. The verte of the parabola is the -intercept. Thus, the onl solution is. Eercises Solve each equation b graphing.. + 7 + = 0. - - = 0. - + 5 = 0 f() f() f() -8-8 - -8 - -, -, - no real roots Chapter 9 Glencoe Algebra Chapter 9 A Glencoe Algebra

Answers (Lesson 9-) Lesson 9- NAME DATE PERID 9- Stud Guide and Intervention (continued) Solving Quadratic Equations b Graphing Estimate Solutions The roots of a quadratic equation ma not be integers. If eact roots cannot be found, the can be estimated b finding the consecutive integers between which the roots lie. Eample Solve + 6 + 6 = 0 b graphing. If integral roots cannot be found, estimate the roots b stating the consecutive integers between which the roots lie. Graph the related function f() = + 6 + 6. f() -5 - - - - - - - Notice that the value of the function changes from negative to positive between the -values of -5 and - and between - and -. f() The -intercepts of the graph are between -5 and - and between - and -. So one root is between -5 and -, and the other root is between - and -. Eercises Solve each equation b graphing. If integral roots cannot be found, estimate the roots to the nearest tenth.. + 7 + 9 = 0. - - = 0. - + 6 = 0 f() f() f() -6 < < -5, - < < -, no real roots - < < - < <. - - = 0 5. - + = 0 6. - - = 0 f() f() f() - < < 0, 0 < <, - < < -, < < 5 < < < < Chapter 9 Glencoe Algebra Answers NAME DATE PERID 9- Skills Practice Solving Quadratic Equations b Graphing Solve each equation b graphing.. - + = 0 Ø. c + 6c + 8 = 0 -, - f() f(c) c. a - a = -. n - 7n = -0, 5 f(a) f(n) n a Solve each equation b graphing. If integral roots cannot be found, estimate the roots to the nearest tenth. 5. p + p + = 0 6. + - = 0 f(p) f() p -., -0.6 -.,. 7. d + 6d = - 8. h + = h f(d) f(h) d h -5.5, -0.6 0.,.8 Chapter 9 Glencoe Algebra Chapter 9 A5 Glencoe Algebra

Lesson 9- NAME DATE PERID 9- Practice Solving Quadratic Equations b Graphing Solve each equation b graphing.. - 5 + 6 = 0,. w + 6w + 9 = 0 -. b - b + = 0 Ø f() f(w) f(b) w b Solve each equation b graphing. If integral roots cannot be found, estimate the roots to the nearest tenth.. p + p = 5. m + 5 = 0m 6. v + 8v = -7 f(p) f(m) f(v) p m v -.6, 0.6 0.6,. -.7, -. 7. NUMBER THERY Two numbers have a sum of and a product of -8. The quadratic equation -n + n + 8 = 0 can be used to determine the two numbers. f(n) a. Graph the related function f(n) = -n + n + 8 and determine its -intercepts. -, b. What are the two numbers? - and n 8. DESIGN A footbridge is suspended from a parabolic support. The function h() = - 5 + 9 represents the height in feet of the support above the walkwa, where = 0 represents the midpoint of the bridge. 6 h() 9. Graph the function and determine its -intercepts. -5, 5 - -6 6-6 0. What is the length of the walkwa between the two supports? 0 ft - Chapter 9 Glencoe Algebra NAME DATE PERID. FARMING In order for Ra to decide how much fertilizer to appl to his corn crop this ear, he reviews records from previous ears. He finds that his crop ield depends on the amount of fertilizer he applies to his fields according to the equation = - + +. Graph the function, and find the point at which Ra gets the highest ield possible. 6 0 8 6 (,6) 5. WRAPPING PAPER Can a rectangular piece of wrapping paper with an area of 8 square inches have a perimeter of 60 inches? (Hint: Let length = 0 w.) Eplain. Solving the equation (0 - w)w = 8 gives w = or 7. A in. b 7 in. sheet of paper would work. 5. ENGINEERING The shape of a satellite dish is often parabolic because of the reflective qualities of parabolas. Suppose a particular satellite dish is modeled b the following equation. 0.5 = + a. Approimate the solution b graphing. - and. LIGHT Azha and Jerem hold a flashlight so that the light falls on a piece of graph paper in the shape of a parabola. Azha and Jerem sketch the shape of the parabola and find that the equation = - - 0 matches the shape of the light beam. Determine the roots of the function. - and 5. FRAMING A rectangular photograph is 7 inches long and 6 inches wide. The photograph is framed using material inches wide. If the area of the frame and photograph combined is 56 square inches, what is the width of the framing material? in. 7 in. Photograph 6 in. Frame Chapter 9 5 Glencoe Algebra Answers (Lesson 9-) 9- Word Problem Practice Solving Quadratic Equations b Graphing --- - - - BA b. n the coordinate plane above, translate the parabola so that there is onl one root. Label this curve A. See students work. c. Translate the parabola so that there are no roots. Label this curve B. See students work. Chapter 9 A6 Glencoe Algebra

Answers (Lesson 9- and Lesson 9-) Lesson 9- NAME DATE PERID 9- Enrichment Parabolas Through Three Given Points If ou know two points on a straight line, ou can find the equation of the line. To find the equation of a parabola, ou need three points on the curve. Here is how to approimate an equation of the parabola through the points (0, -), (, 0), and (5, ). Use the general equation = a + b + c. B substituting the given values for and, ou get three equations. (0, -): - = c (, 0): 0 = 9a + b + c (5, ): = 5a + 5b + c First, substitute - for c in the second and third equations. Then solve those two equations as ou would an sstem of two equations. Multipl the second equation b 5 and the third equation b -. 0 = 9a + b - Multipl b 5. 0 = 5a + 5b - 0 0 = 5a + 5b - Multipl b -. -6 = -75a - 5b + 6-6 = -0a - a = 5 for a in either the second or third equation. NAME DATE PERID 9- Stud Guide and Intervention Transformations of Quadratic Functions Translations A translation is a change in the position of a figure either up, down, or diagonal. When a constant c is added to or subtracted from the parent function, the resulting function f() ± c is a translation of the graph up or down. The graph of f() = + c translates the graph of f() = verticall. f() If c > 0, the graph of f() = is translated c units up. If c < 0, the graph of f() = is translated c units down. c >0 c =0 c <0 Eample f() =. Describe how the graph of each function is related to the graph of a. g() = + The function can be written as f() = + c. The value of c is, and > 0. Therefore, the graph of g() = + is a translation of the graph of f() = up units. g() b. h() = - The function can be written as f() = + c. The value of c is, and < 0. Therefore, the graph of g() = is a translation of the graph of f() = down units. f() f() h() Eercises Describe how the graph of each function is related to the graph of f() =.. g() = +. h() = 6. g() = Translation of Translation of Translation of f() = up unit. f() = down 6 units. f() = down unit.. h() = 0 + 5. g() = + 6. h() = - + Translation of Translation of Translation of f() = up 0 units. f() = down units. f() = down unit. 7. g() = + 8 9 8. h() = 0. 9. g() = Translation of Translation of Translation of f() = up 8 f() = down 0. unit. f() = down units. 9 unit. f() = Chapter 9 7 Glencoe Algebra To find b, substitute 5 0 = 9 ( 5) + b - b = 7 5 The equation of a parabola through the three points is = 5 + 7 5 -. Find the equation of a parabola through each set of three points.. (, 5), (0, 6), (, ). (-5, 0), (0, 0), (8, 00) = - - + 6 = 5 6 + 5 6. (, -), (0, ), (, -). (, ), (6, 0), (0, 0) = - - + = - 5 + 8 5 5. (, ), (5, -), (0, -) 6. (0, ), (, 0), (-, ) = - 9 0 + 8 0 - = - 8 - + Chapter 9 6 Glencoe Algebra Answers Chapter 9 A7 Glencoe Algebra

Answers (Lesson 9-) Lesson 9- NAME DATE PERID 9- Stud Guide and Intervention (continued) Transformations of Quadratic Functions Dilations and Reflections A dilation is a transformation that makes the graph narrower or wider than the parent graph. A reflection flips a figure over the - or -ais. The graph of f() = a stretches or verticall compresses the graph of f() =. a> a= If a >, the graph of f() = is stretched verticall. If 0 < a <, the graph of f() = is compressed verticall. 0<a< The graph of the function f() flips the graph of f() = across the -ais. f()= The graph of the function f( ) flips the graph of f() = across the -ais. f()=- Eample f() =. Describe how the graph of each function is related to the graph of a. g() = The function can be written as f() = a where a =. Because a >, the graph of = is the graph of = that is stretched verticall. f()= b. h() = - - The negative sign causes a reflection across the -ais. Then a dilation occurs in which a = and a translation in which c =. So the graph of = - - is reflected across the -ais, dilated wider than the graph of f()= f() =, and translated down units. f() h() Eercises Describe how the graph of each function is related to the graph of f() =.. h() = -5. g() = - +. g() = - - Compression of Translation of = Dilation of = = narrower than reflected over the wider than the graph of the graph of f() = -ais and up unit. f() = reflected over reflected over the the -ais translated -ais. down unit. Chapter 9 8 Glencoe Algebra NAME DATE PERID 9- Skills Practice Transformations of Quadratic Functions Describe how the graph of each function is related to the graph of f() =.. g() = +. h() = - +. g() = - 8 Translation of = Translation of = Translation of = up units down unit down 8 units. h() = 7 5. g() = 6. h() = -6 Compression of = 5 Dilation of = wider Compression of narrower than the graph than the graph of = narrower than of f() = f() = the graph of f() = reflected over the -ais 7. g() = - + 8. h() = 5-9. g() = + Reflection of = Dilation of = wider Compression of over the -ais than the graph of f() = = narrower translated up units reflected over the than the graph of -ais, and translated f() = translated up 5 units up unit Match each equation to its graph. 0. = - B A. C.. = - D. = - + C C. D.. = - + A Chapter 9 9 Glencoe Algebra Chapter 9 A8 Glencoe Algebra

Answers (Lesson 9-) Lesson 9- NAME DATE PERID 9- Practice Transformations of Quadratic Functions Describe how the graph of each function is related to the graph of f() =.. g() = 0 +. h() = - 5 +. g() = 9 - Translation of = Translation of = Reflection of up 0 units. down = across the 5 unit. = -ais translated up 9 units.. h() = + 5. g() = - - 6. h() = - Compression of = Dilation of = wider Compression of = narrower than the than the graph of narrower than the graph of f() = f() =, reflected over graph of f() =, translated up units. the -ais, translated reflected over the down unit. -ais translated up units. Match each equation to its graph. A. B. C. 7. = - - A 8. = - + C 9. = + B List the functions in order from the most verticall stretched to the least verticall stretched graph. 0. f() =, g() =, h() = -. f() =, g() = - f(), g(), h() h(), f(), g() 6, h() =. PARACHUTING Two parachutists jump from two different planes as part of an aerial show. The height h of the first parachutist in feet after t seconds is modeled b the function h = -6t + 5000. The height h of the second parachutist in feet after t seconds is modeled b the function h = -6t + 000. h = t a. What is the parent function of the two functions given? b. Describe the transformations needed to obtain the graph of h from the parent function. Compression of = narrower than the graph of f() =, reflected over the -ais, translated up 5000 units. c. Which parachutist will reach the ground first? the second parachutist Chapter 9 0 Glencoe Algebra Answers NAME DATE PERID 9- Word Problem Practice. SPRINGS The potential energ stored in a spring is given b the function U s = k where k is a constant known as the spring constant, and is the distance the spring is stretched or compressed from its initial position. Eplain how the graph of the function for a spring where k = newtons/meter differs from the graph of the function for a spring where k = 0 newtons/meter. The graph of U s = (0) is a compression of the graph.. CYLINDERS The volume of a clinder is given b the equation V = πr l, where r is the radius and is the length. A poster compan wants to increase the volume of its -foot long shipping tube b -cubic feet without increasing the length. Eplain how the graph of the original tube differs from the graph of the newl redesigned tube. The graph of the new tube is a translation down units.. PHYSICS A ball is dropped from a height of 0 feet. The function h = -6t + 0 models the height of the ball in feet after t seconds. Graph the function and compare this graph to the graph of its parent function. 8 5 9 Height (ft) 6 0 Transformations of Quadratic Functions 0. 0. 0.6 0.8.0...6 Time (s). ACCELERATIN The distance d in feet a car accelerating at 6 ft/s travels after t seconds is modeled b the function d = t. Suppose that at the same time the first car begins accelerating, a second car begins accelerating at ft/s eactl 00 feet down the road from the first car. The distance traveled b second car is modeled b the function d = t + 00. a. Graph and label each function on the same coordinate plane. 800 700 600 500 00 00 00 00 0 Distance (ft) 6 8 0 6 Time (s) d= t d= t + 00 b. Eplain how each graph is related to the graph of f() =. d= t is a compression of d=t ; d= t + 00 is a compression of d=t translated up 00 units (feet). c. After how man seconds will the first car pass the second car? 0 seconds h = -6t + 0 is a compression of h = t reflected over the -ais and translated up 0 units (feet). Chapter 9 Glencoe Algebra Chapter 9 A9 Glencoe Algebra

Answers (Lesson 9- and Lesson 9-) Lesson X- 9- NAME DATE PERID 9- Enrichment Translating Quadratic Graphs When a figure is moved to a new position without undergoing an rotation, then the figure is said to have been translated to that position. The graph of a quadratic equation in the form = ( - b) + c is a translation of the graph of =. Start with =. Slide to the right units. = ( - ) Then slide up units. = ( - ) + These equations have the form = ( - b). Graph each equation.. = ( - ). = ( - ). = ( + ) These equations have the form = ( - b) + c. Graph each equation.. = ( - ) + 5. = ( - ) + 6. = ( + ) - Chapter 9 Glencoe Algebra NAME DATE PERID 9- Stud Guide and Intervention Solving Quadratic Equations b Completing the Square Complete the Square Perfect square trinomials can be solved quickl b taking the square root of both sides of the equation. A quadratic equation that is not in perfect square form can be made into a perfect square b a method called completing the square. Completing the Square To complete the square for an quadratic equation of the form + b: Step Find one-half of b, the coefficient of. Step Square the result in Step. Step Add the result of Step to + b. + b + ( b ) = ( + b ) Eample Find the value of c that makes + + c a perfect square trinomial. Step Find of. = Step Square the result of Step. = Step Add the result of Step to +. + +. Thus, c =. Notice that + + equals ( + ) Eercises Find the value of c that makes each trinomial a perfect square.. + 0 + c 5. + + c 9. - + c. - 8 + c 6 5. + 5 + c 5 6. + 9 + c 8 7. - + c 9 8. - 5 + c 5 9. + 8 + c 96 0. + + c Chapter 9 Glencoe Algebra Chapter 9 A0 Glencoe Algebra

Answers (Lesson 9-) Lesson 9- NAME DATE PERID 9- Stud Guide and Intervention (continued) Solving Quadratic Equations b Completing the Square Solve b Completing the Square Since few quadratic epressions are perfect square trinomials, the method of completing the square can be used to solve some quadratic equations. Use the following steps to complete the square for a quadratic epression of the form a + b. Step Find b. Step Find ( b ). Step Add ( b ) to a + b. Eample Solve + 6 + = 0 b completing the square. + 6 + = 0 riginal equation + 6 + - = 0 - Subtract from each side. + 6 = 7 Simplif. + 6 + 9 = 7 + 9 Since ( 6 = 9, add 9 to each ) side. ( + ) = 6 Factor + 6 + 9. + = ± Take the square root of each side. = - + Simplif. = - + or = - - = = -7 The solution set is {-7, }. Eercises Solve each equation b completing the square. Round to the nearest tenth if necessar.. - + = 0. + 0 = -9. - 8-9 = 0, -, -9 -, 9. - 6 = 6 5. - - 5 = 0 6. - = 9 -, 8 -, 5-0.7,.7 7. + 8 = 0 8. = + 9. + 0 + = -8-0, -0.,. -9, - 0. - = 5. = +. - 8 = -7-0., 5. -,, 7. + 0 =. - 8 = 9 5. + 6 = -6 -, -, 9 -.9, -. 6. = + 7. + + = 8 8. = 0 + -, -, -, Chapter 9 Glencoe Algebra Answers NAME DATE PERID 9- Skills Practice Solving Quadratic Equations b Completing the Square Find the value of c that makes each trinomial a perfect square.. + 6 + c 9. + + c. - + c 9. - + c 5. - 8 + c 8 6. + 0 + c 00 7. + 5 + c 6.5 8. - 70 + c 5 9. - + c 0.5 0. + 9 + c 0.5 Solve each equation b completing the square. Round to the nearest tenth if necessar.. + - = 0, -6. - 8 + 5 = 0, 5. + 6 = 7-7,. - = 5 -, 5 5. - + 0 = 6, 6. + + = 0 -, - 7. - + = 0 0.,.7 8. - 6 + = 0 0.8, 5. 9. - 8 + 0 = 0.6, 6. 0. - = 5 -.,.. + 0 = - -9.9, -0.. 0.5 + 8 = -7-5., -0.9 Chapter 9 5 Glencoe Algebra Chapter 9 A Glencoe Algebra

Lesson 9- NAME DATE PERID 9- Practice Solving Quadratic Equations b Completing the Square Find the value of c that makes each trinomial a perfect square.. - + c. + 8 + c 96. + 0 + c 00. + + c 9 5. - 9 + c 8 6. - + c Solve each equation b completing the square. Round to the nearest tenth if necessar. 7. - + = 0 8. + = 9. - 0 + 56 = -5, -,, 7 0. + 8 + 9 = 0. - 0 + 6 = -7. + 8 + 50 = 9-6.6, -..5, 8.5-5., -.7. + 5 - = 0. - 7 = 5. 0.9 + 5. - = 0-5., 0. -., 8. -6, 6. 0. + 0.8 = 0. 7. - - 0 = 0 8. + - = 0 -., 0. -, 5-7.,. 9. NUMBER THERY The product of two consecutive even integers is 78. Find the integers. 6, 8 0. BUSINESS Jaime owns a business making decorative boes to store jewelr, mementos, and other valuables. The function = + 50 + 800 models the profit that Jaime has made in month for the first two ears of his business. a. Write an equation representing the month in which Jaime s profit is $00. + 50 + 800 = 00 b. Use completing the square to find out in which month Jaime s profit is $00. the tenth month. PHYSICS From a height of 56 feet above a lake on a cliff, Mikaela throws a rock out over the lake. The height H of the rock t seconds after Mikaela throws it is represented b the equation H = -6t + t + 56. To the nearest tenth of a second, how long does it take the rock to reach the lake below? (Hint: Replace H with 0.) 5. s Chapter 9 6 Glencoe Algebra NAME DATE PERID 9- Word Problem Practice Solving Quadratic Equations b Completing the Square. INTERIR DESIGN Modular carpeting is installed in small pieces rather than as a large roll so that onl a few pieces need to be replaced if a small area is damaged. Suppose the room shown in the diagram below is being fitted with modular carpeting. Complete the square to determine the number of ft b ft squares of carpeting needed to finish the room. Fill in the missing terms in the corresponding equation below. 5; 5. FRGS A frog sitting on a stump feet high hops off and lands on the ground. During its leap, its height h (in feet) is given b h = -0.5d + d +, where d is the distance from the base of the stump. How far is the frog from the base of the stump when it landed on the ground? + 0 or about 5.6 ft 5. GARDENING Peg is planning a rectangular vegetable garden using 50 feet of fencing material. She onl needs to fence three sides of the garden since one side borders an eisting fence. + 0 + = ( + ). FALLING BJECTS Keisha throws a rock down an old well. The distance d (in feet) the rock falls after t seconds can be represented b the equation d = 6t + 6t. If the water in the well is 80 feet below ground, how man seconds will it take for the rock to hit the water? second. MARS n Mars, the gravit acting on an object is less than that on Earth. n Earth, a golf ball hit with an initial upward velocit of 6 meters per second will reach a maimum height of about.5 meters. The height h of an object on Mars that leaves the ground with an initial velocit of 6 meters per second is given b the equation h = -.9t + 6t. Find the maimum height if the same golf ball is hit on Mars. Round our answer to the nearest tenth. 88.9 m Chapter 9 7 Glencoe Algebra Answers (Lesson 9-) a. Let = the width of the rectangle. Write an epression to represent the area of the garden if she uses all the fencing material. (50 - ) b. Find the verte of the equation and identif it as a maimum or a minimum. (6.5, 78.5); maimum c. Interpret the verte of the equation in terms of the situation. If the short side is 6.5 ft and the other side 5 ft, the garden will be the largest possible, with an area of 78.5 ft. Chapter 9 A Glencoe Algebra

Answers (Lesson 9- and Lesson 9-5) Lesson 9-5 NAME DATE PERID 9- Enrichment Factoring Quartic Polnomials Completing the square is a useful tool for factoring and solving quadratic epressions. You can utilize a similar technique to factor simple quartic polnomials of the form + c. Eample Factor the quartic polnomial + 6. Step Find the value of the middle term needed to complete the square. This value is ( 6 ) ( ), or 6. Step Rewrite the original polnomial in factorable form: ( + 6 + ( 6 ) ) - 6. Step Factor the polnomials: ( + 8) - (). Step Rewrite using the difference of two squares: ( + 8 + ) ( + 8 - ) The factored form of + 6 is ( + + 8) ( - + 8). This could then be factored further, if needed, to find the solutions to a quartic equation. Eercises Factor each quartic polnomial.. +. + ( + + )( - + ) ( + 6 + 8)( - 6 + 8). + 500. + 960 ( + 0 + 50)( - 0 + 50) ( + + 98)( - + 98) 5. + 0 6. + 58 ( + 8 + )( - 8 + ) ( + + 7)( - + 7) 7. + 8 8. + 9 ( + + )( - + ) ( + 6 + 9)( - 6 + 9) 9. + 0. 8 + 6,8 ( + 6 + )( - 6 + ) ( + 6 + 8)( - 6 + 8). Factor + c to come up with a general rule for factoring quartic polnomials. ( + c + c ) + ( + c + c ) Chapter 9 8 Glencoe Algebra Answers NAME DATE PERID 9-5 Stud Guide and Intervention Solving Quadratic Equations b Using the Quadratic Formula Quadratic Formula To solve the standard form of the quadratic equation, a + b + c = 0, use the Quadratic Formula. Quadratic Formula the formula = -b ± b - ac a that gives the solutions of a + b + c = 0, where a 0 Eample Eample Solve + = b using the Quadratic Formula. Rewrite the equation in standard form. + = riginal equation + - = - Subtract from each side. + - = 0 Simplif. Now let a =, b =, and c = - in the Quadratic Formula. -b ± b - ac = a a = = - ± () - ()(-) () - + 6 = - + or = - + = = - The solution set is {-, }. Solve - 6 - = 0 b using the Quadratic Formula. Round to the nearest tenth if necessar. For this equation a =, b = -6, and c = -. = -b ± b - ac a = 6 ± (-6) - ()(-) () = 6 ± = 6 ± or = 6-6. -0. The solution set is {-0., 6.}. Eercises Solve each equation b using the Quadratic Formula. Round to the nearest tenth if necessar.. - + = 0,. - 8 = -6. 6-8 = -. + 5 = 6-6, 5. + = 8 -, 6. 8-8 - 5 = 0-0.,. 7. - + 9 = 7, 8. + 6 = 5 -.7, 0.7 9. 8 + - 5 = 0-5 6, 8 0. 8 - = -,. + 5 = 8 -.6,.. 8 + 9 - = 0 -.5, 0.. + 9 + = 0 -, -. 8 + 7 + = 0 -, - 8 Chapter 9 9 Glencoe Algebra Chapter 9 A Glencoe Algebra

Answers (Lesson 9-5) Lesson 9-5 NAME DATE PERID 9-5 Stud Guide and Intervention (continued) Solving Quadratic Equations b Using the Quadratic Formula The Discriminant In the Quadratic Formula, = -b ± b - ac a, the epression under the radical sign, b - ac, is called the discriminant. The discriminant can be used to determine the number of real roots for a quadratic equation. Case : b - ac < 0 no real roots Case : b - ac = 0 one real root Case : b - ac > 0 two real roots Eample State the value of the discriminant for each equation. Then determine the number of real solutions of the equation. a. + 5 = Write the equation in standard form. + 5 = riginal equation + 5 - = - Subtract from each side. + 5 - = 0 Simplif. Now find the discriminant. b - ac = (5) - ()(-) = 7 Since the discriminant is positive, the equation has two real roots. b. + = - + = - riginal equation + + = - + Add to each side. + + = 0 Simplif. Find the discriminant. b - ac = () - ()() = - Since the discriminant is negative, the equation has no real roots. Eercises State the value of the discriminant for each equation. Then determine the number of real solutions of the equation.. + - = 0. - 7-8 = 0. - 0-9 = 0 0, real roots 5, real solutions 7, real solutions. = + 5. - = 0 6. 6-0 + 0 = 0 65, real solutions 89, real solutions -0, no real solutions 7. - 0 = - 8. 6 = - - 0 9. 9-8 + 9 = 0 6, real solutions -89, no real solutions 0, real solution 0. + 9 = -6. 9 = 8. 6 + 6 + = 0-96, no real solutions 96, real solutions 0, real solutions. 8 + 9 =. - + = 5. - 8 = - 5, real solutions 0, real solution 56, real solutions Chapter 9 0 Glencoe Algebra NAME DATE PERID 9-5 Skills Practice Solving Quadratic Equations b Using the Quadratic Formula Solve each equation b using the Quadratic Formula. Round to the nearest tenth if necessar.. - 9 = 0-7, 7. - - 0 = 0 -, 5. - 5-6 = 0 -, 9. + + 0 = 0-6, -5 5. - 7 = - 0.5, 6.5 6. + = - -.7, -0. 7. - 9 + = 0 8. + 6 + = 0-5., -0.6 9. + 5-7 = 0-0. - = -, 0.,. + 5 + = 0. + 7 = 9 -,. + - = 0 -., 0.7. - 7-6 = 0 -, State the value of the discriminant for each equation. Then determine the number of real solutions of the equation. 5. + + = 0 6. + + = 0 ; real solutions 0; real solution 7. - + 0 = 0 8. - 6 + 7 = 0 -; no real solutions 8; real solutions 9. - - 7 = 0 0. - 0 + 5 = 0 ; real solutions 0; real solution. + 5-8 = 0. + 6 + = 0 89; real solutions -60; no real solutions. - + 0 = 0. + 7 + = 0-6; no real solutions ; real solutions Chapter 9 Glencoe Algebra Chapter 9 A Glencoe Algebra

Lesson 9-5 NAME DATE PERID 9-5 Practice Solving Quadratic Equations b Using the Quadratic Formula Solve each equation b using the Quadratic Formula. Round to the nearest tenth if necessar.. + - = 0 -,. + 8 + 7 = 0-7, -. - + 6 = 0. - 6 + 7 = 0.6,. 5. + 9-5 = 0-5, 6. + + 0 = 0-5, - 7. - 9 = - 8. - 5 = - 9. + = -, 9., 0. - = -8 -.8, 0.. + 7 = 5 -,..6 + +.5 = 0..5 + -.5 = 0. + + = 0 5. - = -., 0. -, - -0., 0.6 State the value of the discriminant for each equation. Then determine the number of real solutions of the equation. 6. + 8 + 6 = 0 7. + + = 0 8. + = -7 0; real solution -9; no real solutions 88; real solutions 9. + 5 = -0 0. + 9 =. - =.5-5; no real solution 0; real solutions 6; real solutions..5 + - 0.5 = 0. - = -. = - - ; real solutions -; no real solutions 0; real solution 5. CNSTRUCTIN A roofer tosses a piece of roofing tile from a roof onto the ground 0 feet below. He tosses the tile with an initial downward velocit of 0 feet per second. a. Write an equation to find how long it takes the tile to hit the ground. Use the model for vertical motion, H = -6t + vt + h, where H is the height of an object after t seconds, v is the initial velocit, and h is the initial height. (Hint: Since the object is thrown down, the initial velocit is negative.) H = -6t - 0t + 0 b. How long does it take the tile to hit the ground? about. s 6. PHYSICS Lupe tosses a ball up to Quen, waiting at a third-stor window, with an initial velocit of 0 feet per second. She releases the ball from a height of 6 feet. The equation h = -6t + 0t + 6 represents the height h of the ball after t seconds. If the ball must reach a height of 5 feet for Quen to catch it, does the ball reach Quen? Eplain. (Hint: Substitute 5 for h and use the discriminant.) No; the discriminant, -6, is negative, so there is no solution. Chapter 9 Glencoe Algebra Answers NAME DATE PERID Word Problem Practice Solving Quadratic Equations b Using the Quadratic Formula. BUSINESS Tana runs a catering business. Based on her records, her weekl profit can be approimated b the function f() = + - 7, where is the number of meals she caters. If f() is negative, it means that the business has lost mone. What is the least number of meals that Tana needs to cater in order to have a profit? 6 meals. CRAFTS Madeln cut a 60-inch pipe cleaner into two unequal pieces, and then she used each piece to make a square. The sum of the areas of the squares was 7 square inches. Let = the length of one piece. Write and solve an equation to represent the situation and find the lengths of the two original pieces. ( 60 - ) + ( ) = 7 in. and 6 in.. AERNAUTICS At liftoff, the space shuttle Discover has a constant acceleration of 6. feet per second squared and an initial velocit of feet per second due to the rotation of Earth. The distance Discover has traveled t seconds after liftoff is given b the equation d(t) = t + 8.t. How long after liftoff has Discover traveled 0,000 feet? Round our answer to the nearest tenth. 5.8 seconds 5. SITE DESIGN The town of Smallport plans to build a new water treatment plant on a rectangular piece of land 75 ards wide and 00 ards long. The buildings and facilities need to cover an area of 0,000 square ards. The town s zoning board wants the site designer to allow as much room as possible between each edge of the site and the buildings and facilities. Let represent the width of the border. 00 d 75 d Buildings and Facilities Border. ARCHITECTURE The Golden Ratio appears in the design of the Greek Parthenon because the width and height of the façade are related b the equation W + H W = W. If the height of a H model of the Parthenon is 6 inches, what is its width? Round our answer to the nearest tenth. 5.9 in. Chapter 9 Glencoe Algebra Answers (Lesson 9-5) 9-5 a. Use an equation similar to A = l w to represent the situation. 0,000 = (00 - )(75 - ) b. Write the equation in standard quadratic form. - 550 + 5000 = 0; 9.8 and 7.7 c. What should be the width of the border? Round our answer to the nearest tenth. 9.79 d Chapter 9 A5 Glencoe Algebra

Lesson 9-6 NAME DATE PERID 9-5 Enrichment Golden Rectangles A golden rectangle has the propert that its sides satisf the following proportion. a + b a = a Two quadratic equations can be written from the proportion. These are sometimes called golden quadratic equations. b b a. In the proportion, let a =. Use. Solve the equation in Eercise for b. cross-multiplication to write a quadratic equation. - + b = 5 b + b - = 0. In the proportion, let b =. Write. Solve the equation in Eercise for a. a quadratic equation in a. a - a - = 0 a = + 5 5. Eplain wh ( 5 + ) and ( 5 - ) are called golden ratios. The are the ratios of the sides in a golden rectangle. The first is the ratio of the long side to the short side; the second is short side: long side. Another propert of golden rectangles is that a square drawn inside a golden rectangle creates another, smaller golden rectangle. C In the design at the right, opposite vertices of each square have been connected with quarters of circles. For eample, the arc from point B to point C is created b putting the point of a compass at point A. The radius of the arc is the length BA. B A 6. n a separate sheet of paper, draw a larger version of the design. Start with a golden rectangle with a long side of 0 inches. The short side should be about 6 6 inches. Chapter 9 Glencoe Algebra NAME DATE PERID 9-6 Stud Guide and Intervention Eponential Functions Graph Eponential Functions Eponential Function a function defined b an equation of the form = a b, where a 0, b > 0, and b You can use values of to find ordered pairs that satisf an eponential function. Then ou can use the ordered pairs to graph the function. Eample Eample Graph =. Find the -intercept and state the domain and range. Graph = ( ). Use the graph to approimate the value of ( ) -0.5 - - 6 9-0 - 8 0 9 6 The -intercept is. The domain is all real numbers, and the range is all positive numbers. The value of ( ) -0.5 is about. Eercises. Graph = 0.. Find the -intercept. Then use the graph to approimate the value of 0. -.5. Use a calculator to confirm the value. ; about 6 Graph each function. Find the -intercept and state the domain and range.. = +. = ( ) +. = ( ) - - Chapter 9 5 Glencoe Algebra. Answers (Lesson 9-5 and Lesson 9-6) Chapter 9 A6 Glencoe Algebra

Answers (Lesson 9-6) Lesson 9-6 NAME DATE PERID 9-6 Stud Guide and Intervention (continued) Eponential Functions Identif Eponential Behavior It is sometimes useful to know if a set of data is eponential. ne wa to tell is to observe the shape of the graph. Another wa is to observe the pattern in the set of data. Eample Determine whether the set of data shown below displas eponential behavior. Write es or no. Eplain wh or wh not. 0 6 8 0 6 6 8 Method : Look for a Pattern The domain values increase b regular intervals of, while the range values have a common factor of. Since the domain values increase b regular intervals and the range values have a common factor, the data are probabl eponential. Method : Graph the Data 8 The graph shows rapidl decreasing values of as increases. This is characteristic of eponential behavior. Eercises Determine whether the set of data shown below displas eponential behavior. Write es or no. Eplain wh or wh not.. 0 5 0 5 0. 0 9 7 8 No; the domain values are at regular Yes; the domain values are at intervals, and the range values have regular intervals, and the range a common difference 5. values have a common factor.. - 5 6 8. - 0 Yes; the domain values are at No; the domain values are at regular intervals, and the range regular intervals, but the range values have a common factor. values do not change. 5. -5 0 5 0 0.5 0.5 0.5 6. 0 Yes; the domain values are at Yes; the domain values are at regular intervals, and the range regular intervals, and the range values have a common factor 0.5. values have a common factor. 9 7 8 Chapter 9 6 Glencoe Algebra Answers NAME DATE PERID 9-6 Skills Practice Eponential Functions Graph each function. Find the -intercept, and state the domain and range. Then use the graph to determine the approimate value of the given epression. Use a calculator to confirm the value.. = ;.. = ( ) ; ( ) -.6 ; D = all reals, R = { > 0};.9 ; D = all reals, R = { > 0}; 5.8 Graph each function. Find the -intercept, and state the domain and range.. = ( ). = + ; D = all reals, R = { > 0} ; D = all reals, R = { > } Determine whether the set of data shown below displas eponential behavior. Write es or no. Eplain wh or wh not. 5. - - - 0 9 5 8 6. 0 5 0 5 0 0 5.5 No; the domain values are at Yes; the domain values are at regular intervals and the range regular intervals and the range values have a common values have a common factor 0.5. difference. 7. 8 6 0 0 80 60 8. 50 0 0-0 90 70 50 0 Yes; the domain values are at No; the domain values are at regular intervals and the range regular intervals and the range values have a common factor. values have a common difference 0. Chapter 9 7 Glencoe Algebra Chapter 9 A7 Glencoe Algebra

Answers (Lesson 9-6) Lesson 9-6 NAME DATE PERID 9-6 Practice Eponential Functions Graph each function. Find the -intercept and state the domain and range. Then use the graph to determine the approimate value of the given epression. Use a calculator to confirm the value.. = ( 0) ; ( 0) -0.5. = ;.9. = ( ) ; ( ) -. ; D = {all real numbers}; R = { >0}; ; D = {all real numbers}; R = { >0}; 8. ; D = {all real numbers}; R = { > 0}; 7.0 Graph each function. Find the -intercept, and state the domain and range.. = ( ) + 5. = ( - ) 6. = 0.5( - ) 5; D = {all real numbers}; R = { >}; 0; D = {all real numbers}; R = { >-} -; D = {all real numbers}; R = { >.5} Determine whether the set of data shown below displas eponential behavior. Write es or no. Eplain wh or wh not. 7. 5 8 8 0 0 7.5 8. 8 5 0 6 9 Yes; the domain values are at No; the domain values are at regular intervals and the range regular intervals and the range values have a common values have a common factor 0.5. difference 7. 9. LEARNING Ms. Klemperer told her English class that each week students tend to forget one sith of the vocabular words the learned the previous week. Suppose a student learns 60 words. The number of words remembered can be described b the function W() = 60 ( 5 6), where is the number of weeks that pass. How man words will the student remember after weeks? about 5 0. BILGY Suppose a certain cell reproduces itself in four hours. If a lab researcher begins with 50 cells, how man cells will there be after one da, two das, and three das? (Hint: Use the eponential function = 50( ).) 00 cells; 0,800 cells;,07,00 cells Chapter 9 8 Glencoe Algebra NAME DATE PERID 9-6 Word Problem Practice Eponential Functions. WASTE Suppose the waste generated b nonreccled paper and cardboard products is approimated b the following function. = 000() 0. Sketch the eponential function on the coordinate grid below. 50 50 850 550 50 950 650 50. DEPRECIATIN The value of Roce Compan s computer equipment is decreasing in value according to the following function. = 000(0.87) In the equation, is the number of ears that have elapsed since the equipment was purchased and is in dollars. What was the value 5 ears after it was purchased? Round our answer to the nearest dollar. $99 ----. MNEY Tatana s grandfather gave her one penn on the da she was born. He plans to double the amount he gives her ever da. Estimate how much she will receive from her grandfather on the th da of her life. about $0 5. METERLGY The atmospheric pressure (in millibars) at a given altitude, in meters, can be approimated b the following function. The function is valid for values of between 0 and 0,000. f () = 08(.000) - a. What is the pressure at sea level? 08 millibars. PICTURE FRAMES Since a picture frame includes a border, the picture must be smaller in area than the entire frame. The table shows the relationship between picture area and frame length for a particular line of frames. Is this an eponential relationship? Eplain. No; there is no common factor between the picture areas. Side Picture Length Area (in.) (in ) 5 6 6 7 0 8 0 9 b. The McDonald bservator in Teas is at an altitude of 000 meters. What is the approimate atmospheric pressure there? 79 millibars c. As altitude increases, what happens to atmospheric pressure? It decreases. Chapter 9 9 Glencoe Algebra Chapter 9 A8 Glencoe Algebra

NAME DATE PERID Enrichment Rational Eponents You have developed the following properties of powers when a is a positive real number and m and p are integers. a m a p = a m + p (ab) m = a m b m a 0 = (a m ) p = a mp a m = a a m - p a -m = p a m Eponents need not be restricted to integers. We can define rational eponents so that operations involving them will be governed b the properties for integer eponents. ( a ) = a = a ( a ) = a ( a n ) n = a n n = a a squared is a. a cubed is a. a n to the n power is a. a is a square root of a. a is a cube root of a. a n is an nth root of a. a = a a = a a n = n a Now let us investigate the meaning of a m p. a m p = a m p = ( a m ) p = p a m a m p = a p m = ( a p ) m = ( p a ) m Therefore, a m p = p a m or ( p a ) m. Write a in eponential form. a 5 = 5 a Write a 5 in radical form. a = a Find a a Chapter 9 0 Glencoe Algebra Answers. Answers (Lesson 9-6 and Lesson 9-7) Lesson 9-7 9-6 Eample Eample a a Eample = a - = a 6 - a 6 = a 6 or 6 Write each epression in radical form.. b b. c c. (c ) c Write each epression in eponential form.. b b 5. a ( a ) = a 6. b b Perform the operation indicated. Answers should show positive eponents onl. 7. ( a b ) a 6 b 8. -8 a a 0. a a a. ( a b - - a ) - b 6 a 9. ( b b - ). - a b 7 b 0 (5 a b - ) -0 a 5 6 b NAME DATE PERID 9-7 Stud Guide and Intervention Growth and Deca Eponential Growth Population increases and growth of monetar investments are eamples of eponential growth. This means that an initial amount increases at a stead rate over time. The general equation for eponential growth is = a( + r) t. represents the final amount. Eponential Growth a represents the initial amount. r represents the rate of change epressed as a decimal. t represents time. Eample Eample PPULATIN The population of Johnson Cit in 000 was 5,000. Since then, the population has grown at an average rate of.% each ear. a. Write an equation to represent the population of Johnson Cit since 000. The rate.% can be written as 0.0. = a( + r) t = 5,000( + 0.0) t = 5,000(.0) t b. According to the equation, what will the population of Johnson Cit be in the ear 00? In 00, t will equal 00-000 or 0. Substitute 0 for t in the equation from part a. = 5,000(.0) 0 t = 0,56 In 00, the population of Johnson Cit will be about,56. INVESTMENT The Garcias have $,000 in a savings account. The bank pas.5% interest on savings accounts, compounded monthl. Find the balance in ears. The rate.5% can be written as 0.05. The special equation for compound interest is A = P ( + r n ) nt, where A represents the balance, P is the initial amount, r represents the annual rate epressed as a decimal, n represents the number of times the interest is compounded each ear, and t represents the number of ears the mone is invested. A = P ( + r n ) nt A =,000( + 0.05 ) 6 A,000(.009) 6 A,8.09 In three ears, the balance of the account will be $,6.9. Eercises. PPULATIN The population of the United. INVESTMENT Determine the States has been increasing at an average amount of an investment of $500 if it annual rate of 0.9%. If the population of the is invested at in interest rate of 5.5% United States was about 0,6,000 in the compounded monthl for ears. ear 008, predict the U.S. population in the $08.78 ear 0. about 08,85,85. PPULATIN It is estimated that the. INVESTMENT Determine the population of the world is increasing at an amount of an investment of $00,000 average annual rate of.%. If the population if it is invested at an interest rate of of the world was about 6,6,000,000 in the 5.% compounded quarterl for ear 008, predict the world population in the ears. ear 05. about 7,08,88,769 $85,888.87 Chapter 9 Glencoe Algebra Chapter 9 A9 Glencoe Algebra