3.1 Graph Quadratic Functions
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1 3. Graph Quadratic Functions in Standard Form Georgia Performance Standard(s) MMA3b, MMA3c Goal p Use intervals of increase and decrease to understand average rates of change of quadratic functions. Your Notes VOCABULARY Quadratic function Parabola Verte Ais of smmetr Minimum and maimum value Etrema 56 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.
2 Your Notes Eample Graph 5. Compare the graph with the graph of 5. Identif the domain and range.. Make a table of values for 5. 0 Graph a function of the form 5 a c. Plot the points from the table Draw a smooth through the points. 4. Compare the graphs. Both graphs have the same. However, the graph of 5 opens and is than the graph of 5. Also, its verte is units higher. 5. Identif the domain and range. The domain is and the range is. Eample Graph Graph a function of the form 5 a b c. Because a 0, the parabola opens.. Find the verte. First, calculate the -coordinate. 5 b } a 5 5 Then find the -coordinate. 5 5 The verte is (, ). Plot this point. 3. Draw the ais of smmetr Identif the -intercept c, which is. Plot the point (0, ). Then reflect this point in the ais of smmetr to plot another point (4, ). 5. Evaluate the function for Plot the point (, ) and its reflection (3, ) in the ais of smmetr. 6. Draw a parabola through the plotted points. Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 57
3 Your Notes Checkpoint Complete the following eercises.. Graph 5 }. Identif the domain and range.. Graph the function. Label the verte and ais of smmetr. 5 4 Eample 3 Find the minimum or maimum value Tell whether the function 53 6 has a minimum value or a maimum value. Then find the minimum or maimum value. Solution Because a 0, the function has a value. To find it, calculate the coordinates of the verte. 5 b } a The maimum value is 5. Checkpoint Complete the following eercise. 3. Tell whether the function has a minimum or maimum value. Then find the minimum or maimum value. 58 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.
4 Your Notes Eample 4 Find the maimum value of a quadratic function Revenue A bike shop sells about 8 bikes each da when the charge $4 per bike. For each $ decrease in price, the sell about 3 more bikes each da. How can the bike shop maimize dail revenue? Solution. Define the variables. Let represent the price reduction and R() represent the dail revenue.. Write a verbal model. Then write and simplif a quadratic function. Revenue (dollars) 5 Price (dollars/bike) p Number of bikes R() 5 ( ) p ( 3) R() 5 R() 5 3. Find the coordinates (, R()) of the verte. 5 b } a 5 5 Find -coordinate. R( ) 53( ) 08( ) 756 Evaluate R( ). The verte is (, ). The shop should reduce the price b to maimize dail revenue. Checkpoint Complete the following eercise. 4. In Eample 4, what is the maimum dail revenue? Homework Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 59
5 Name Date LESSON 3. Practice Cop and complete the table of values for the function ????? 0????? } 4. 5 } ????? ????? For the following functions (a) tell whether the graph opens up or opens down, (b) find the verte, and (c) find the ais of smmetr Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.
6 Name Date LESSON 3. Practice continued Match the equation with its graph } 5 A. B. C. Graph the function. Label the verte and ais of smmetr Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 6
7 Name Date LESSON 3. Practice continued Graph the function. Label the verte and ais of smmetr } 3. 5 } 3. 5 } 4 } 3. Wrist Watch Brand X wrist watches at a department store are selling for $50 at a rate of 45 per month. The marketing department determined that for ever $ decrease in price, 3 more watches would be sold per month. Write a quadratic equation in standard form that models the revenue R from watch sales. How can the store maimize monthl revenue? 4. Motocross The path that a motocross dirt bike rider follows during a jump is given b where is the horizontal distance (in feet) from the edge of the ramp and is the height (in feet). What is the maimum height of the rider during the jump? Height (feet) Horizontal distance (feet) 6 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.
8 3. Graph Quadratic Functions in Verte or Intercept Form Georgia Performance Standard(s) MMA3a, MMA3c Your Notes Goal p Graph quadratic functions in verte form or intercept form. VOCABULARY Verte form Intercept form Eample Graph a quadratic function in verte form Graph 5 } ( ).. Identif the constants a 5, h 5, and k 5. Because a > 0, the parabola opens.. Plot the verte (h, k) 5 (, ) and draw the ais of smmetr at Evaluate the function for two values of. 5 : : 5 6 Plot the points (, ) and (3, ) and their reflections in the ais of smmetr. 4. Draw a parabola through the plotted points. Checkpoint Graph the function. Label the verte and ais of smmetr.. 5( 3) 4 Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 63
9 Your Notes Eample Graph 5( )( 5).. Identif the -intercepts. Because p 5 and q 5, the -intercepts occur at the points (, 0) and (, 0).. Find the coordinates of the verte. 5 p q } 5 Graph a quadratic function in intercept form So, the verte is (, ). 3. Draw a parabola through the verte and the points where the -intercepts occur. Checkpoint Graph the function. Label the verte, ais of smmetr, and -intercepts.. 5 ( 4)( ) 3. 5( )( 3) 64 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.
10 Your Notes Eample 3 Use a quadratic model in verte form Towers Two generator towers are designed with an electric cable that connects them. The ends of the cable are the same height above the ground. The cable can be modeled b ( 400) 5 } where is the horizontal distance (in feet) from the left tower and is the corresponding height (in feet) of the cable. Find the distance between the towers. Solution The verte of the parabola is (, ). The cable's lowest point is feet from either tower. The distance between the towers is d 5 ( ) 5 feet. Checkpoint Complete the following eercise. 4. Suppose in Eample 3, the cable is modeled b ( 500) 5 } Find the distance between the towers. Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 65
11 Your Notes Eample 4 a. 5 3( )( 5) Change quadratic functions to standard form b. 55( ) 8 Solution a. 5 3( )( 5) Original function 5 3 Multipl using FOIL. 5 3 Combine like terms. 5 Distributive propert b. 55( ) 8 Original function 5 5( )( ) 8 Rewrite ( ). 5 5( ) 8 Multipl using FOIL. 5 5( ) 8 Combine like terms. 5 8 Distributive propert 5 Combine like terms. Checkpoint Write the quadratic function in standard form ( 3) ( 7)( 6) Homework 66 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.
12 Name Date LESSON 3. Practice Match the equation with its graph.. 5 ( ). 5 ( )( 4) 3. 5 ( ) 3 A. B. C. 4 Graph the function. Label the verte and ais of smmetr ( ) 5. 5 ( 3) 6. 5 ( ) 7. 5 ( ) ( ) 9. 5 ( 3) 3 Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 67
13 Name Date LESSON 3. Practice continued Graph the function. Label the verte, ais of smmetr, and -intercepts ( )( 5). 5 ( )( ). 5 ( 6)( ) 3. 5 ( 3)( ) 4. 5 ( )( ) ( )( 4) 3 3 Write the quadratic function in standard form ( ) 7. 5 ( 3) ( ) ( 3)( ) 0. 5 ( )( 4). 5 3( )( 3) 68 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.
14 Name Date LESSON 3. Practice continued Find the minimum value or the maimum value of the function.. 5 ( 3) 3. 5 ( ) ( ) ( 3)( ) 6. 5 ( )( 5) ( 3)( ) In Eercises 8 and 9, use the following information. Golf The flight of a particular golf shot can be modeled b the function ( 80) where is the horizontal distance (in ards) from the impact point and is the height (in ards). The graph is shown below. Height (ards) Horizontal distance (ards) 8. How man ards awa from the impact point does the golf ball land? 9. What is the maimum height in ards of the golf shot? Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 69
15 3.3 Interpret Rates of Change of Quadratic Functions Georgia Performance Standard(s) MMA3c Goal p Use intervals of increase and decrease to understand average rates of change of quadratic functions. Your Notes Eample Identif intervals of increase and decrease Graph the function 5. Identif the intervals over which the graph increases and decreases. Solution You can see from the graph that as ou move from left to right the value of the function on the left side of the verte and on the right side of the verte. The -coordinate of the verte is 5 b } a 5 5. The graph over the interval > and over the interval <. Checkpoint Graph the function. Identif the intervals over which the graph increases and decreases.. 5 } 70 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.
16 Your Notes Eample Calculate the average rate of change Calculate the average rate of change of the function 5 on the interval 0. Solution Find the two points on the graph of the function that correspond to the endpoints of the interval. The average rate of change is the slope of the line that passes through these two points. 5 ( ) 5 ( ) The points are and. The average rate of change is: r Eample 3 Compare average rates of change Compare the average rates of change of 5 and 5 on 0. Solution The average rate of change of 5 is the slope of the line, which is. The points and correspond to the endpoints of the interval for 5. The average rate of change of 5 on 0 is r The average rate of change of the quadratic function is times as great as the average rate of change of the linear function on the interval 0. Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 7
17 Your Notes Checkpoint Complete the following eercises.. Calculate the average rate of change of 5 3 on the interval. 3. Compare the average rates of change of 5 } 4 and 5 on 4. Eample 4 Solve a real world problem Baseball The path of a baseball thrown at an angle of 408 can be modeled b where is the horizontal distance (in feet) from the release point and is the corresponding height (in feet). Find the interval on which the height is increasing. Solution The height of the baseball will be increasing from the release point until it reaches its maimum height at the verte. The -coordinate of the verte is 5. So, the height will be increasing on the interval. Homework Checkpoint For the quadratic model in Eample 4, find the average rate of change on the given interval Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.
18 Name Date LESSON 3.3 Practice Graph the function. Identif the intervals over which the graph increases and decreases ( ) 3. 5 ( ) ( 4)( ) Calculate the average rate of change of the function on the given interval. 5. 5, 6. 5 }, , , 3 Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 73
19 Name Date LESSON 3.3 Practice continued Classif the given interval as an interval of increase or decrease Compare the average rates of change of the functions on the given interval and 5 7 on and 5 } on 0 4. Basketball The path of a basketball after being thrown can be modeled b the function where is the horizontal distance (in feet) from where the ball was thrown and is the corresponding height (in feet). Find the interval on which the height is increasing. What is the average rate of change on this interval? 74 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.
20 3.4 Solve b c 5 0 b Factoring Georgia Performance Standard(s) MMA3c, MMA4b Your Notes Goal p Use factoring to solve quadratic equations. VOCABULARY Monomial Binomial Trinomial Quadratic equation Root of an equation Zero of a function SPECIAL FACTORING PATTERNS Pattern Name Difference of a b 5 ( )( ) Two Squares 4 5 ( )( ) Perfect Square a ab b 5 ( ) Trinomial ( 3) Perfect Square a ab b 5 ( ) Trinomial ( ) Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 75
21 Your Notes Eample Factor trinomials of the form b c Factor the epression 7 8. Solution You want ( m)( n) where mn 5 and m n 5. Factors of 8 (m, n),, Sum of factors (m n) Factors of 8 (m, n),, Sum of factors (m n) Notice that m 5 and n 5. So, ( )( ). Eample Factor with special patterns Factor the epression. a. 5 5 Difference of two squares 5 ( )( ) b. m m Perfect square trinomial 5 m (m)( ) 5 ( ) Checkpoint Factor the epression. If it cannot be factored, sa so Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.
22 Your Notes Eample 3 Find the roots of an equation Find the roots of the equation Solution Original equation ( )( ) 5 0 Factor. 5 0 or 5 0 Zero product propert 5 or 5 Solve for. The roots are and. Eample 4 Use a quadratic equation as a model Patio A rectangular patio measures 0 feet b 30 feet. B adding feet to the width and feet to the length, the area is doubled. Find the new dimensions of the patio. Solution New area 5 New width p New length ( )( ) 5 ( ) p ( ) 5 Multipl using FOIL. 0 5 Write in standard form. 0 5 ( )( ) Factor. 5 0 or 5 0 Zero product propert 5 or 5 Solve for. Reject the negative value. The patio's length and width should each be increased b feet. The new dimensions are feet b feet. Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 77
23 Your Notes Checkpoint Complete the following eercises. 3. Find the roots of Rework Eample 4 where a rectangular patio measures 6 feet b 4 feet. Eample 5 Find the zeros of a quadratic function Find the zeros of the function b rewriting the function in intercept form. Solution Write original equation. 5 ( )( ) Factor. The zeros of the function are and. CHECK Graph The graph passes through (, 0) and (, 0). Checkpoint Complete the following eercise. Homework 5. Find the zeros of the function b rewriting the function in intercept form. 78 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.
24 Name Date LESSON 3.4 Practice Factor the epression. If the epression cannot be factored, sa so Solve the equation Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 79
25 Name Date LESSON 3.4 Practice continued Find the zeros of the function b rewriting the function in intercept form f() g() h() Find the value of. 37. Area of the rectangle Area of the rectangle Hopscotch The communit plaground has a hopscotch pad that is 8 feet longer than it is wide. The total area of the pad is 48 square feet. What are the dimensions of the hopscotch pad? 80 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.
26 3.5 Solve a b c b Factoring Georgia Performance Standard(s) MMA4b Goal p Use factoring to solve equations of the form a b c 5 0. Eample Factor a b c where c < 0 Your Notes Factor 3. You want 3 5 (k m)(l n) where k and l are factors of and m and n are factors of. Because mn 0, m and n have signs. k, l m, n (k m)(l n) a b c 3,, ( )( ) 3,, ( )( ) 3,, ( )( ) 3,, ( )( ) The correct factorization is 3 5 ( )( ). Checkpoint Factor the epression. If it cannot be factored, sa so Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 8
27 Your Notes Eample Factor with special patterns Factor the epression. a ( ) Difference of 5 ( )( ) two squares b Perfect square trinomial 5 ( ) ( )( ) 5 ( ) c. 4m 0m 4 5( ) Factor monomial first. 5( )( ) Factor. Checkpoint Factor the epression Eample 3 Solve quadratic equations a. 5 0 Original equation ( )( ) 5 0 Factor. 5 0 or 5 0 Zero product propert 5 or 5 Solve for. b. 4r 8r 4 5 6r Original equation 5 0 Standard form 5 0 Divide each side b. 5 0 Factor. 5 0 Zero product propert r 5 Solve for r. 8 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.
28 Your Notes Checkpoint Solve the equation z 3z 5 5z 4 Eample 4 Use a quadratic equation as a model Mirror The area of a mirror is 0 square inches and the length is 3 more inches than times the width. Find the length of the mirror. Solution Area of mirror (square inches) 5 Width of mirror (inches) p Length of mirror (inches) 5 p ( ) 0 5 Write in standard form. 0 5 ( )( ) Factor. 5 0 or 5 0 Zero product propert 5 or 5 Solve for. Reject the negative value. The length of the mirror is inches or inches. Checkpoint Complete the following eercise. 7. Rework Eample 4 where the length is 6 inches more than the width. Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 83
29 Your Notes Eample 5 Solve a multi-step problem Game A store sells about 80 board games per week when it charges $0 per game. For each decrease of $, the store sells 0 more board games. How much should the store charge to maimize revenue? What is the maimum weekl revenue? Solution. Define the variables. Let represent the number of $ price decreases and R() represent the weekl revenue.. Write a verbal model. Then write and simplif a quadratic function. Weekl sales (dollars) 5 Number of board games sold p Price of board game (dollars/game) 5 p 5 3. Identif the zeros and find their average. The zeros are and. The average of the zeros is. To maimize revenue, the store should charge. 4. Identif the maimum weekl revenue.. The maimum weekl revenue is. Checkpoint Complete the following eercise. Homework 8. In Eample 5, if the store were to onl decrease the price b $, what would be their weekl revenue? 84 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.
30 Name Date LESSON 3.5 Practice Factor the epression. If the epression cannot be factored, sa so Solve the equation Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 85
31 Name Date LESSON 3.5 Practice continued Find the zeros of the function b rewriting the function in intercept form f() g() Find the value of. 34. Area of the square Area of the rectangle Pool A pool deck of uniform width is going to be built around a rectangular pool that is 0 feet long and 5 feet wide. After the deck is built, a total of 44 square feet will be occupied. How wide is the deck encompassing the pool? ft 5 ft ft ft 0 ft ft 86 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.
32 3.6 Solve Quadratic Equations b Finding Square Roots Georgia Performance Standard(s) MMA4b Your Notes Goal p Solve quadratic equations b finding square roots. VOCABULARY Rationalizing the denominator Conjugates RATIONALIZING THE DENOMINATOR Form of denominator Ï } b a Ï } b a Ï } b Multipl numerator and denominator b Ï } b a Ï } b a Ï } b Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 87
33 Your Notes Eample Simplif (a) Î } 7 }3 and (b) Rationalize denominators of fractions 4 } 5 Ï } 3. a. Î } 7 }3 5 p 5 b. 4 } 5 Ï } } 5 Ï } 3 p 5 5 Checkpoint Simplif the epression.. Ï } 5 p Ï } 0. Î } 9 } Eample Finding solutions of a quadratic equation Find the solutions of } 4 ( 6) 5 8. } 4 ( 6) 5 8 Write original equation. 5 Multipl each side b. 5 Take square roots of each side. 5 Add to each side. 5 Simplif. The solutions are and. 88 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.
34 Your Notes Eample 3 Model a dropped object with a quadratic function Water Balloon A water balloon is dropped from a window 59 feet above the sidewalk. How long does it take for the water balloon to hit the sidewalk? Solution h 5 6t h 0 Write height function. 5 6t Substitute for h and for h t Subtract from each side. 5 t Divide each side b. 5 t Take square roots of each side. ø t Use a calculator. Reject the negative solution,, because time must be positive. The water balloon will fall for about seconds before it hits the ground. Checkpoint Complete the following eercises. 3. Solve the equation Homework 4. In Eample 4, suppose that the water balloon is dropped from a height of 7 feet. How long does it take for the balloon to hit the sidewalk? Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 89
35 Name Date LESSON 3.6 Practice Simplif the epression.. Ï } 98. Ï } 9 p 3 Ï } 7 3. Ï } p Ï } 7 4. Î } 5 } 6 5. Î } 49 } 9 6. Î } 00 } 5 7. Î } 7 } 8. Î } 45 } 3 9. Î } 4 } 5 p 3 Î} } 5 0. } 3 Ï } 3. } 5 Ï } 6. Ï } 5 } 5 Ï } 5 90 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.
36 Name Date LESSON 3.6 Practice continued Solve the equation } } When an object is dropped, its height h (in feet) above the ground can be modeled b h 5 6t h 0 where h 0 is the object s initial height (in feet). Find the time it takes an object to hit the ground when it is dropped from a height of h 0 feet. 8. h h h h h h Multiple Choice What are all the solutions to ? A. Ï }, Ï } B. 3, 3 C. 3 Ï }, 3 Ï } D. 3 Ï } 35. New Car From 980 to 000, the average cost of a new car C (in dollars) can be approimated b the model C t 600 where t is the number of ears since 980. During which ear was the average cost of a new car $5,05? Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 9
37 3.7 Complete the Square Georgia Performance Standard(s) MMA3a, MMA4b Your Notes Goal p Solve quadratic equations b completing the square. VOCABULARY Completing the square COMPLETING THE SQUARE Words To complete the square for the epression b, add. Algebra b 5 } b } b 5 Eample Make a perfect square trinomial Find the value of c that makes c a perfect square trinomial. Then write the epression as the square of a binomial. Solution. Find half the coefficient of.. Square the result of Step Replace c with the result of Step. The trinomial c is a perfect square when c 5. Then 5 ( )( ) 5 ( ). 5 9 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.
38 Your Notes Checkpoint Find the value of c that makes the epression a perfect square trinomial. Then write the epression as the square of a binomial.. 4 c. 0 c Eample Solve a b c 5 0 when a 5 Solve b completing the square. Solution Write original equation. 5 Write left side in the form b. 5 Complete the square. 5 Write left side as a binomial squared. 5 Take square roots of each side. 5 Solve for. 5 Simplif. The solutions are and. Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 93
39 Your Notes Eample 3 Solve a b c 5 0 when a Þ Solve b completing the square. Solution Write original equation. 5 Divide each side b the coefficient of. 5 Write left side in the form b. 5 Complete the square. 5 Write left side as binomial squared. 5 Take square roots of each side. 5 Write in terms of the imaginar unit i. The solutions are and. Checkpoint Solve the equation b completing the square Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.
40 Your Notes Eample 4 Find the maimum value of a quadratic equation Profit A store's profit is modeled b P 5 (00 8)(35 ). Rewrite in verte form to find the number of units that maimizes the profit. Solution P 5 P 5 P 5 P 5 P 5 P 5 The verte is maimizes P is. Write original function. Use FOIL. Combine like terms. Prepare to complete the square. Add and subtract. Write a perfect square trinomial as the square of a binomial., so the number of units that Checkpoint Complete the following eercise. 5. Rework Eample 4 where P 5 (00 5)(50 ). Homework Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 95
41 Name Date LESSON 3.7 Practice Solve the equation b finding square roots } Find the value of c that makes the epression a perfect square trinomial. Then write the epression as the square of a binomial c. c. 8 c 3. 4 c 4. 4 c 5. 5 c 6. c 7. 7 c 96 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.
42 Name Date LESSON 3.7 Practice continued Solve the equation b completing the square Write the quadratic function in verte form. Then identif the verte Find the value of. 30. Area of rectangle Area of rectangle Area of triangle Area of triangle Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 97
43 3.8 Use the Quadratic Formula and the Discriminant Georgia Performance Standard(s) MMA4b, MMA4c Your Notes Goal p Solve quadratic equations using the quadratic formula. VOCABULARY Quadratic formula Discriminant THE QUADRATIC FORMULA Let a, b, and c be real numbers such that a Þ 0. The solutions of the quadratic equation a b c are: 5 6 Ï }} 4 }} Eample Solve an equation with two real solutions Solve Original equation Standard form Ï }} 4 Quadratic formula }} 6 Ï }} 4 }}} a 5, b 5, c 5 5 Simplif. The solutions are 5 ø and 5 ø. 98 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.
44 Your Notes Eample Solve an equation with one real solution Solve Solution Ï }} 4 }}} The solution is. Original equation a 5, b 5, c 5 Simplif. Simplif. Eample 3 Solve an equation with imaginar solutions Solve 5 5. Solution 5 5 Original equation 5 0 Standard form 5 6 Ï }} 4 }}} a 5, b 5, c 5 5 Simplif. 5 5 Simplif. The solutions are and. Rewrite using the imaginar unit i. Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 99
45 Your Notes Checkpoint Use the quadratic formula to solve the equation USING THE DISCRIMINANT OF a b c 5 0 When b 4ac > 0, the equation has. The graph has -intercepts. When b 4ac 5 0, the equation has. The graph has -intercept. When b 4ac < 0, the equation has. The graph has -intercepts. 00 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.
46 Your Notes Eample 4 Use the discriminant Find the discriminant of the quadratic equation and give the number and tpe of solutions of the equation. a b c Discriminant Solution(s) b 4ac 5 b 6 Ï} b 4ac }} a a. b. c. Checkpoint Find the discriminant of the quadratic equation and give the number and tpe of solutions of the equation. Then solve the equation Homework Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 0
47 Name Date LESSON 3.8 Practice Write the equation in standard form. Identif a, b, and c Find the discriminant and use it to determine if the solution has one real, two real, or two imaginar solution(s) Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.
48 Name Date LESSON 3.8 Practice continued Use the quadratic formula to solve the equation Solve the equation using the quadratic formula. Then solve the equation b factoring to check our solution(s) Find the value of. 34. Area of rectangle Area of parallelogram Horseshoes A contestant tosses a horseshoe from one pit to another with an initial vertical velocit of 50 feet per second. The horseshoe is released 3 feet above the ground. Use the model h 5 6t 50t 3 where h is the height (in feet) and t is the time (in seconds) to tell how long the horseshoe was in the air. Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 03
49 3.9 Graph and Solve Quadratic Inequalities Georgia Performance Standard(s) MMA4d Your Notes Goal p Graph and solve quadratic inequalities. VOCABULARY Quadratic inequalit in two variables Quadratic inequalit in one variable Eample Use a quadratic inequalit in real life Bridge A suspension bridge can support a maimum weight W (in pounds) if W 3000d where d is the diameter of the steel cables (in inches) used to suspend the bridge. Graph the inequalit. Solution Graph W d for nonnegative values of d. Because the inequalit smbol is, make the parabola. Test the point (5, 0,000) which is below the parabola. W 3000d? 3000( ) W (5, 0,000) d Because (5, 0,000) the parabola. a solution, shade the region 04 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.
50 Your Notes Checkpoint Complete the following eercise.. Rework Eample given the new inequalit W 500d. W d Eample Graph a sstem of quadratic inequalities Graph the sstem of quadratic inequalities. > Inequalit 3 4 Inequalit Solution. Graph >. The graph is the region (but not including) the parabola 5.. Graph 3 4. The graph is the region and including the parabola Identif the region where the two graphs overlap. This region is the graph of the sstem. Checkpoint Complete the following eercise.. Graph the sstem. < 3 3 Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 05
51 Your Notes Eample 3 Solve 3 4. Solve a quadratic inequalit using a table Rewrite the inequalit as Then make a table of values Notice that when the values of are between and, inclusive. The solution of the inequalit is. Eample 4 Solve The solution consists of the -values for which the graph of lies the -ais. Find the graph s -intercepts b letting 5 0 and using to solve for Solve a quadratic inequalit b graphing 5 ø or ø Sketch a parabola that opens and has and as -intercepts. The graph lies the -ais to the left of (and including) 5 and to the right of (and including) 5. The solution is approimatel. 06 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.
52 Your Notes Checkpoint Complete the following eercises. 3. Solve the quadratic inequalit 3 using a table. 4. Solve the quadratic inequalit 3 5 b graphing. Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 07
53 Your Notes Eample 5 Solve. Solve a quadratic inequalit algebraicall First, write and solve the equation obtained b replacing with. Write corresponding equation. Write in standard form. Factor. Zero product propert The numbers are the critical -values of the inequalit. Plot on a number line, using dots. The critical -values partition the number line into three intervals. Test an -value in each interval to see if it satisfies the inequalit. Test 5 : Test 5 : Test 5 : The solution is. Checkpoint Complete the following eercise. 5. Solve the inequalit algebraicall. 6 Homework 08 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.
54 Name Date LESSON 3.9 Practice Determine whether the ordered pair is a solution of the inequalit.. < 4, (0, 0). > 4, (, 7) 3. 5, (, ) 4. 3, (3, 4) 5. < 6, (6, 5) 6. > 8, (, 4) Match the inequalit with its graph <. >. > 4 3 A. B. C. D. E. F. Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 09
55 Name Date LESSON 3.9 Practice continued Graph the inequalit < < 4 7. > > Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.
56 Name Date LESSON 3.9 Practice continued Match the sstem of inequalities with its graph.. > 3. > 4. < < < > A. B. C. Graph the sstem of inequalities. 5. > < < 3 > In Eercises 8 and 9, use the following information. Construction A paint can is dropped from the top of a 500 foot tall building being constructed. The height of the paint can can be modeled b h 5 6t 500 where h is the height (in feet) and t is the time (in seconds). A platform is 00 feet from the ground. 8. Write an inequalit that shows when the paint can is above the platform. 9. For what values of t is the paint can above the platform? Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics
57 Words to Review Give an eample of the vocabular word. Quadratic function Parabola Verte Ais of smmetr Minimum and maimum values Verte form Intercept form Monomial Binomial Trinomial Quadratic Equation Root of an equation Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.
58 Zero of a function Rationalizing the denominator Conjugates Completing the square Quadratic formula Discriminant Quadratic inequalit in two variables Quadratic inequalit in one variable Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 3
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