Read To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions Find these vocabular words in Lesson 5-1 and the Multilingual Glossar. Vocabular quadratic function parabola verte of a parabola verte form Translating Quadratic Functions Using the graph of f ( ) as a guide, describe the transformations, and then graph the function. g () ( 3 ) 1 f ( h) ( h ) represents the general form for a horizontal shift. If h 0 the graph moves left and if h 0 the graph moves. f () k k represents the general form for a vertical shift. If k is negative the graph is shifted down and if k is positive the graph is shifted. g () ( 3 ) 1 ( ( 3) ) 1 Rewrite to identif h and k. Because h, the graph is translated 3 units left and since k 1, the graph is translated unit down. Complete the table of values and graph. f () ( 3 ) 1 (, f ()) 5 f ( 5) ( 5 3 ) 1 3 ( 5, 3) 6 8 6 f ( ) ( 3 ) 1 (, ) 3 f ( 3) ( 3 3 ) 1 ( 3, ) f ( ) ( 3 ) 1 (, ) 1 f ( 1) ( 3 ) 1 ( 1, ) Writing Transformed Quadratic Functions Use the description to write the quadratic function in verte form: f ( ) is verticall stretched b a factor of 3 and translated units left. The form of a quadratic function is f () a( h ) k. The a indicates a across the -ais and/or a vertical or compression. The h represents a translation and indicates a vertical translation. Vertical stretch b 3: means 3. Translated units left means h. Substitute to write the transformed function. g () a( h ) k g () ( ) 0 g () ( ) 73 Holt Algebra
Read To Go On? Skills Intervention 5- Properties of Quadratic Functions in Standard Form Find these vocabular words in Lesson 5- and the Multilingual Glossar. Vocabular ais of smmetr standard form minimum value maimum value Graphing Quadratic Functions in Standard Form For the function g () 3, (a) determine whether the graph opens upward or downward, (b) find the ais of smmetr, (c) find the verte, (d) find the -intercept, and (e) graph the function. g () 3 a. What is the standard form for a quadratic equation? f () c. The parabola opens if a 0 and downward if a 0. Since a equals 1 in the given function, the graph will open. b. The ais of smmetr is given b a. What does b equal in the function g() 3? b Substitute to find. a 1 ( 1) The ais of smmetr is the line. c. The verte lies on the ais of smmetr, so the -coordinate is. The f ( 1) 3 1 3 The verte is at (, ). is the value of the function at this -value, or f ( 1). d. The -intercept is the c value of the function. In the function, g () 3 c therefore, the -intercept is. e. Graph the function. Step 1 Plot the ais of smmetr. Step Plot the verte. (, ) Step 3 Plot the -intercept. (, ). Step Use the ais of smmetr to find another point on the parabola. Step 5 Connect the points in a smooth curve. 5 5 5 5 7 Holt Algebra
Read To Go On? Problem Solving Intervention 5- Properties of Quadratic Functions in Standard Form When a parabola opens upward, the -value of the verte is a minimum value. When a parabola opens downward, the -value of the verte is the maimum value. In a science eperiment a ball is rolled up a ramp and allowed to roll back down. The equation that models the distance between the ball and the top of the ramp is h() 0.33 0.88 1. Find the minimum distance between the ball and the top of the ramp. Understand the Problem 1. What does the equation model?. What is the given equation? 3. What are ou asked to find? Make a Plan. Where is the minimum distance located in a parabola? 5. What is the equation that models the distance? 6. To find the -value of the verte, use the formula a. Solve 7. What is the a value in the equation h () 0.33 0.88 1? 8. What is the b value in the equation h () 0.33 0.88 1? b 9. Substitute into the formula: 1.5 a ( ) 10. Find the -value of the verte, h(1.5) 0.33(1.5 ) 0.88( ) 1 h(1.5) 11. The minimum distance between the ball and the top of the ramp is in. Look Back 1. Graph h() 0.33 0.88 1 on a graphing calculator. 13. Locate the minimum value of the parabola. Does our answer check? 75 Holt Algebra
Read To Go On? Skills Intervention 5-3 Solving Quadratic Equations b Graphing and Factoring Find these vocabular words in Lesson 5-3 and the Multilingual Glossar. Vocabular zero of a function root of an equation binomial trinomial Finding Zeros b Factoring Find the roots of each equation b factoring. A. 3 10 Set the function equal to 0. 3 0 What are two factors of 10 whose sum equal 3? and Write the equation in factored form: ( )( ) 0 Appl the Zero Product Propert: 0 or 0 5 0 Solve each equation: or 0 Check b substituting each value of into the original equation. 3 10 3 10 5 3( ) 10 ( ) 3( ) 10 5 10 10 B. 6 18 0 10 10 10 10 First, find the Greatest Common Factor of 6 and 18. Factor the equation: 6( ) 0 Appl the Zero Product Propert: 0 or 3 0 6 0 Solve each equation: or 3 0 Check b substituting each value into the original equation. 6 18 0 6 18 0 6( ) 18(0) 0 6( ) 18( 3) 0 0 0 5 0 0 0 0 0 76 Holt Algebra
Read To Go On? Skills Intervention 5- Completing the Square Find this vocabular word in Lesson 5- and the Multilingual Glossar. Solving a Quadratic Equation b Completing the Square Solve the equation 8 11, b completing the square. 8 11 Set up to complete the square. What is the value of b in the equation? What is b? 8 11 Add b to both sides. 8 Simplif. ( ) 5 Factor. Take the square root of both sides. or Solve for. or The solutions of the equation are. Vocabular completing the square Writing a Quadratic Function in Verte Form Write the function f () 8 5, in verte form, and identif its verte. f () ( ) 5 Factor so the coefficient of the -term is 1. f () ( ) 5 Set up to complete the square. What is b? What term did ou factor out so the leading coefficient was 1? You must subtract b from 5. f () 5 f () ( ) 5 8 Simplif. f () ( ) Simplif. f () ( ) Factor. The verte form of a quadratic function is f () a( h ) k. So, what is the value of h? What is the value of k? The verte of the function is at (, ). 77 Holt Algebra
Read To Go On? Skills Intervention 5-5 Comple Numbers and Roots Find these vocabular words in Lesson 5-5 and the Multilingual Glossar. Vocabular imaginar unit imaginar number comple number real part imaginar part comple conjugate Solving a Quadratic Equation with Imaginar Solutions Solve each equation. A. 5 15 5 15 To isolate divide both sides of the equation b. Check Simplif. The opposite of squaring a number is taking the Take the square root of both sides. The square root of a negative number is an i Epress in terms of i. 5 15 ( 5i ) 15 5(5) 15 number. of the number. Substitute the answer back into the original equation. Simplif. 15( ) 15 Substitute i 1. B. m 6m 1 15 15 m 6m 1 Set up to complete the square. What is the value of b? What is b? m 6m 1 Add b to both sides. m 6m 9 Simplif. (m ) Factor. m 3 Take the square root of both sides. m 3 i Simplif, b subtracting from both sides. The two solutions are and. 78 Holt Algebra
Read To Go On? Skills Intervention 5-6 The Quadratic Formula Find this vocabular word in Lesson 5-6 and the Multilingual Glossar. Quadratic Functions with Comple Zeros Find the zeros of the function 5 0 b using the Quadratic Formula. What is the general form for a quadratic equation? Vocabular discriminant What is the quadratic formula? In the equation 5 0 what are a, b, and c? a b and c. Substitute for a, b, and c in the quadratic formula. (1) Simplif. (1) 80 (1) Simplif. 5 5 i What is the square root of a negative number? Now, write in terms of i. i Analzing Quadratic Equations b Using the Discriminant Find the tpe and number of solutions for the equation 8 9 0. The is part of the quadratic formula that can be used to determine the number of real roots of a quadratic equation. If b ac 0, there are two distinct If b ac 0, then there is solutions. distinct real solution. If b ac 0, then there are two distinct nonreal solutions. What are a, b, and c in the given equation? a, b, c The discriminant is b ac. Substitute a, b, and c. ( 8 ) ( )( ) 78 Since b ac 0, the equation has distinct real solution. 79 Holt Algebra
Read To Go On? Quiz 5-1 Using Transformations to Graph Quadratic Functions Using the graph of f( ) ( ) as a guide, describe the transformations, and then graph each function. 1. g () ( 3 ) 3. g () 3( ) 3. g () 1 1 6 6 Use the description to write each quadratic function in verte form.. f () () is verticall stretched b a factor 5. f () () is reflected across the -ais of and translated units left to create g (). and translated 3 units up to create g (). 5- Properties of Quadratic Functions in Standard Form. For each function, (a) determine whether the graph opens upward or downward, (b) find the ais of smmetr, (c) find the verte, (d) find the -intercept, and (e) graph the function. 6. f () 6 5 7. g () 3 8. h() a) a) a) b) b) b) c) c) c) d) d) d) e) e) e) 6 6 80 Holt Algebra
Read To Go On? Quiz continued 9. A soccer ball is kicked and modeled b the function h() 0.0085 0.5 6, where h is the height of the ball in feet and is the horizontal distance in feet that the ball travels. Find the maimum height of the ball to the nearest foot. 5-3 Solving Quadratic Equations b Graphing and Factoring Find the roots of each equation b factoring. 10. 1 0 11. 3 8 1. 5 15 0 5- Completing the Square Solve each equation b completing the square. 13. 6 7 1. 6 5 15. 8 7 Write each function in verte form, and identif its verte. 16. f () 18 77 17. g () 6 13 18. h () 3 6 6 5-5 Comple Numbers and Roots Solve each equation. 19. 3 75 0. 13 1. 9 5-6 The Quadratic Formula Find the zeros of each function b using the Quadratic Formula.. f () ( 8 ) 3 3. g () 3 7. h() 3 8 5 Find the tpe and number of solutions for each equation. 5. 1 6. 9 1 7. 5 8 0 81 Holt Algebra
Read To Go On? Enrichment Equations Quadratic in Form Certain equations that are not quadratic can be thought of in such a wa that the can be solved as quadratic. For eample, because the square of is, the equation 9 8 0 is said to be quadratic in. 9 8 0 ( ) 9( ) 8 0 Think of as ( ). u 9u 8 0 Substitute u for. The equation u 9u 8 0 can be solved b factoring or b the quadratic formula. u 9u 8 0 (u 8)(u 1) 0 u 8 0 or u 1 0 u 8 or u 1 Replace u with and solve these equations: 8 1 8 1 1 The solutions are 1, 1, and. Solve. 1. 6 9 0. 7 1 0 3. 6 19 15 0. 6 17 5 0 5. 13 36 0 6. 6 8 3 7 0 8 Holt Algebra
5B Read To Go On? Skills Intervention 5-7 Solving Quadratic Inequalities Find this vocabular word in Lesson 5-7 and the Multilingual Glossar. Solving Quadratic Inequalities b Using Algebra Solve the inequalit 3 8. 3 8 Write the related equation b replacing the inequalit smbol with the sign. 3 8 Write the equation in standard form, b subtracting from both sides. 3 0 Simplif. This is now the equation in standard form. Factor the equation. What are the factors of 10 that have a sum of 3? Write the equation in factored form. ( )( ) 0 Write the factors. and 0 or 0 Appl the Zero Product Propert. or Solve each equation for. Plot the two solutions on a number line. The critical values divide the number line into 6 5 3 1 0 1 3 5 6 The intervals are,, and. intervals. Determine if the test values make the original inequalit, 3 8, true or false. Tr 6. ( 6 ) 3( 6) 8 True or False? Tr 0. (0 ) 3(0) 8 True or False? Tr. ( ) 3() 8 True or False? Should the circle drawn on 5 be solid or empt? Should the circle drawn on be solid or empt? Shade the solutions on the number line, where the test points made the inequalit true. The solution is or. Vocabular 6 5 3 1 0 1 3 5 6 quadratic inequalities in two variables 83 Holt Algebra
5B Read To Go On? Problem Solving Intervention 5-7 Solving Quadratic Inequalities A profit is made when the revenue from items sold is more than the cost to produce the items. A loss occurs when cost is more than the revenue. A business makes and sells cabinets. The profit that the compan earns for number of cabinets can be modeled b P() 30 750 000. How man cabinets are needed for a profit of at least $000? Understand the Problem 1. How much must the profit be?. Which smbol is needed to represent at least? Make a Plan 3. Write the inequalit to represent this situation. 30 750 000 000 Look Back. Find the critical values b solving the related equation. 30 750 000 000 Write as an equation. 30 750 0 Write the equation in standard form. 10( 00) 0 Factor out 10 to simplif. Substitute a, b, c into the quadratic formula and solve. ( 75) ( ) ( )(00) b b ac a 75 6 5. Plot the -values on the number line and test an -value in each of the three regions formed b the critical -values. or 6 8 10 1 1 16 18 0 Tr 6. Tr 1. Tr 18. 30(6 ) 750(6) 000 0 30 750 000 0 30 750 000 0 580 0 0 0 True or False? True or False? True or False? 6. For a profit of at least $000, from to cabinets must be sold. Look Back 7. Enter 30 750 000 into a graphing calculator, and create a table of values. Does the table show that integer values of between 8 and 17 inclusive results in -values greater than or equal to 000? 8 Holt Algebra
5B Read To Go On? Skills Intervention 5-8 Curve Fitting with Quadratic Models Find these vocabular words in Lesson 5-8 and the Multilingual Glossar. Vocabular quadratic model quadratic regression Writing a Quadratic Function from Data Write a quadratic function that fits the points (0, 1), (, ), and (3, 3). Use each point to write a sstem of equations to find a, b, and c in f () a b c. Substitute and into the general form of a quadratic function. (, ) f () a b c Sstem in a, b, c Equation (0, 1) 1 a(0 ) b(0) c c 1 (, ) a( ) b() c a c (3, 3) 3 a(3 ) b(3) c 3b c 3 3 Substitute c 1 from Equation 1 into both Equations and 3. a b c Equation 9a 3b c 3 Equation 3 a b 9a 3b 3 a b ( 1) 1 Add 1. 9a 3b ( 1) 1 3 Add 1. a b Equation 9a 3b Equation 5 Net solve Equation and Equation 5 for a and b using elimination. Equation a b 8 Multipl b 3. a(3) b(3) 8(3) 1a b Equation 5 9a 3b 15 Multipl b. 9a( ) 3b( ) 15( ) Substitute the solution for a into Equation to find b. a 6b 30 6a 6 a ( ) b 8 b b Subtract from both sides. Divide b. The function is: f () a b c 85 Holt Algebra
5B Read To Go On? Problem Solving Intervention 5-8 Curve Fitting with Quadratic Models A quadratic model is a quadratic function that represents a set of real data. Models are helpful for making estimates. A graphing calculator can help to make predictions from a set of data. Claire is participating in a running club and keeps record of how man miles she runs. The table shows the distances that Claire has run after so man das. Find the quadratic model for the number of miles ran in the amount of das given. Use the model to estimate the number of miles that Claire ran in 5 das. Predict the number of miles she will have run after 55 das. Claire s Running Record Das Miles 10 1.5 0 36.0 30 69.5 0 11.0 50 169.5 Understand the Problem 1. What does the data in the first column represent?. What does the data in the second column represent? 3. In 30 das, how man miles has Claire ran?. According to the chart, in 5 das she should have ran between 36 and miles. Make a Plan 5. On our calculator, which data will ou enter for List 1? 10,,,, 6. On our calculator, which data will ou enter for List? 1.5,,,, Solve 7. Using the quadratic regression feature on our calculator, a 0.06, b, and c 0.. 8. The quadratic model is 0.0 5 0.. 9. Using the table feature, when 5,. (To the nearest hundredth.) 10. Using the table feature, when 55,. (To the nearest hundredth.) Look Back 11. Graph the function model from Eercise 8 on a graphing calculator. Does the model appear to fit the data? 86 Holt Algebra
5B Read To Go On? Skills Intervention 5-9 Operations with Comple Numbers Find these vocabular words in Lesson 5-9 and the Multilingual Glossar. Vocabular comple plane absolute value of a comple number Adding and Subtracting Comple Numbers Add or subtract. Write the result in the form a bi. A. (6 7i ) (5 3i) (6 ) (7i ) Group the real parts and the imaginar parts. 11 i Add. Then, write the result in a bi form. B. (7 i ) (5 3i ) 7 i i Distribute the negative sign. (7 5) ( 3i ) Group the real parts and the imaginar parts. i Add. Then, write the result in a bi form. Multipling Comple Numbers Multipl. Write the result in the form a bi. A. 7i(3 i ) B. (7 i )( 6i ) 8 i Distribute. 1 i i Multipl. 8( ) Use i 1. 1 i Combine like terms. 1i Multipl. 1 1( ) Use i 1. 8 Write in a bi form. 1 Simplif. i Write in a bi form. Evaluate Powers of i. Simplif 7 i 17. 7i i Rewrite as an even power. 7i ( i ) Rewrite as a power of i. A negative number raised to an even power has a 7i ( ) Simplif i 1. i Simplif. solution. 87 Holt Algebra
5B Read To Go On? Quiz 5-7 Solving Quadratic Inequalities Graph each inequalit. 1. 3. 5 6 Solve each inequalit b using tables or graphs. 3. 8. 1 8 Solve each inequalit b algebra. 5. 3 15 3 6. 0 7. The function P() 15 600 1800 models the monthl profit P of a small business, where is the price of an item. For what price of items does the store earn a monthl profit of at least $1950? 5-8 Curve Fitting with Quadratic Models Determine whether each data set could represent a quadratic function. Eplain. 8. 6 7 8 9 10 11 9 5 1 9 9. 6 3 0 1 3 8 6 6 8 88 Holt Algebra
5B Read To Go On? Quiz continued Write a quadratic function that fits each set of points. 10. (0, 10), (, 0), and (3, ) 11. (1, 5), (, 6), and (, ) For Eercises 1 1, use the table of number of recliners produced and the profit made over three months. 1. Use the data to find a quadratic function that describes the profit as a function of number of recliners produced. 13. Use our function to predict the level of production that will maimize the profit. Month Number of Recliners Produced Profit 1 50 $5100 100 $5600 3 150 $1100 1. Use our function to predict the maimum profit, assuming that all business situations sta the same. 5-9 Operations with Comple Numbers Find each absolute value. 15. i 16. 1 5i 17. 3i Perform each indicated operation, and write the result in the form a bi. 18. ( 5i ) (6 3i ) 19. (8 6i ) (3 i ) 0. i(3 5i ) 1. ( 5 i )( i ). (1 i )(1 i ) 3. i 35. 6i 3 3i 5. 3 i 3i 89 Holt Algebra
5B Read To Go On? Enrichment Solving Quadratic Sstems of Inequalities Graphical techniques can be used to solve sstems of quadratic inequalities. When graphing two parabolas, the solution set includes all the points (, ) in both shaded regions. For eample, consider this quadratic inequalit sstem: { The graph of is a parabola opening upward. The graph of is a parabola opening downward. The intersection of the shaded regions shows the solutions. Solve. 1. 3 1. ( 3 ) 3 ( 1 ) 3. 1. 3( 3 ) ( 1 ) 90 Holt Algebra