8.3 Graphing f ( ) = a + b + c Essential Question How can ou find the verte of the graph of f () = a + b + c? Comparing -Intercepts with the Verte Work with a partner. a. Sketch the graphs of = 8 and = 8 + 6. b. What do ou notice about the -coordinate of the verte of each graph? c. Use the graph of = 8 to find its -intercepts. Verif our answer b solving = 8. d. Compare the value of the -coordinate of the verte with the values of the -intercepts. Finding -Intercepts Work with a partner. a. Solve = a + b for b factoring. b. What are the -intercepts of the graph of = a + b? c. Cop and complete the table to verif our answer. CONSTRUCTING VIABLE ARGUMENTS To be proficient in math, ou need to make conjectures and build a logical progression of statements. b a = a + b Deductive Reasoning Work with a partner. Complete the following logical argument. The -intercepts of the graph of = a + b are and b a. The verte of the graph of = a + b occurs when =. The vertices of the graphs of = a + b and = a + b + c have the same -coordinate. The verte of the graph of = a + b + c occurs when =. Communicate Your Answer 4. How can ou find the verte of the graph of f () = a + b + c? 5. Without graphing, find the verte of the graph of f () = 4 + 3. Check our result b graphing. Section 8.3 Graphing f() = a + b + c 43
8.3 Lesson What You Will Learn Core Vocabular maimum value, p. 433 minimum value, p. 433 Previous independent variable dependent variable Graph quadratic functions of the form f () = a + b + c. Find maimum and minimum values of quadratic functions. Graphing f ( ) = a + b + c Core Concept Graphing f ( ) = a + b + c The graph opens up when a >, and the graph opens down when a <. The -intercept is c. The -coordinate of the verte is b a. The ais of smmetr is = b a. (, c) = b a verte f() = a + b + c, where a > Finding the Ais of Smmetr and the Verte Find (a) the ais of smmetr and (b) the verte of the graph of f () = + 8. a. Find the ais of smmetr when a = and b = 8. = b a = 8 () = The ais of smmetr is =. Write the equation for the ais of smmetr. Substitute for a and 8 for b. Simplif. Check b. The ais of smmetr is =, so the -coordinate of the verte is. Use the function to find the -coordinate of the verte. f() = + 8 4 f () = + 8 Write the function. 66 f ( ) = ( ) + 8( ) Substitute for. = 9 Simplif. The verte is (, 9). X=- Y=-9 Monitoring Progress Help in English and Spanish at BigIdeasMath.com Find (a) the ais of smmetr and (b) the verte of the graph of the function.. f () = 3. g() = + 6 + 5 3. h() = + 7 4 43 Chapter 8 Graphing Quadratic Functions
Graphing f () = a + b + c COMMON ERROR Be sure to include the negative sign before the fraction when finding the ais of smmetr. Graph f () = 3 6 + 5. Describe the domain and range. Step Find and graph the ais of smmetr. = b a = ( 6) = Substitute and simplif. (3) Step Find and plot the verte. The ais of smmetr is =, so the -coordinate of the verte is. Use the function to find the -coordinate of the verte. f () = 3 6 + 5 Write the function. f () = 3() 6() + 5 Substitute for. = Simplif. So, the verte is (, ). 4 f() = 3 6 + 5 Step 3 Use the -intercept to find two more points on the graph. Because c = 5, the -intercept is 5. So, (, 5) lies on the graph. Because the ais of smmetr is =, the point (, 5) also lies on the graph. 4 6 Step 4 Draw a smooth curve through the points. REMEMBER The domain is the set of all possible input values of the independent variable. The range is the set of all possible output values of the dependent variable. The domain is all real numbers. The range is. Monitoring Progress Graph the function. Describe the domain and range. Help in English and Spanish at BigIdeasMath.com 4. h() = + 4 + 5. k() = 8 + 7 6. p() = 5 Finding Maimum and Minimum Values Core Concept Maimum and Minimum Values The -coordinate of the verte of the graph of f () = a + b + c is the maimum value of the function when a < or the minimum value of the function when a >. f () = a + b + c, a < f () = a + b + c, a > maimum minimum Section 8.3 Graphing f() = a + b + c 433
Finding a Maimum or Minimum Value Tell whether the function f () = 4 4 9 has a minimum value or a maimum value. Then find the value. For f () = 4 4 9, a = 4 and 4 <. So, the parabola opens down and the function has a maimum value. To find the maimum value, find the -coordinate of the verte. First, find the -coordinate of the verte. Use a = 4 and b = 4. = b a = ( 4) ( 4) = 3 Substitute and simplif. Then evaluate the function when = 3 to find the -coordinate of the verte. f ( 3) = 4( 3) 4( 3) 9 Substitute 3 for. = 7 Simplif. The maimum value is 7. Finding a Minimum Value The suspension cables between the two towers of the Mackinac Bridge in Michigan form a parabola that can be modeled b =.98.37 + 55, where and are measured in feet. What is the height of the cable above the water at its lowest point? 5 5 5 5 5 3 35 434 Chapter 8 Graphing Quadratic Functions The lowest point of the cable is at the verte of the parabola. Find the -coordinate of the verte. Use a =.98 and b =.37. = b a = (.37) 888 Substitute and use a calculator. (.98) Substitute 888 for in the equation to find the -coordinate of the verte. =.98(888).37(888) + 55 3 The cable is about 3 feet above the water at its lowest point. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Tell whether the function has a minimum value or a maimum value. Then find the value. 7. g() = 8 8 + 6 8. h() = 4 + 3 + 9. The cables between the two towers of the Tacoma Narrows Bridge in Washington form a parabola that can be modeled b =.6.46 + 57, where and are measured in feet. What is the height of the cable above the water at its lowest point?
in. 3 4 5 6 Modeling with Mathematics A group of friends is launching water balloons. The function f (t) = 6t + 8t + 5 represents the height (in feet) of the first water balloon t seconds after it is launched. The height of the second water balloon t seconds after it is launched is shown in the graph. Which water balloon went higher? 5 nd balloon MODELING WITH MATHEMATICS Because time cannot be negative, use onl nonnegative values of t. 4 6 t. Understand the Problem You are given a function that represents the height of the first water balloon. The height of the second water balloon is represented graphicall. You need to find and compare the maimum heights of the water balloons.. Make a Plan To compare the maimum heights, represent both functions graphicall. Use a graphing calculator to graph f (t) = 6t + 8t + 5 in an appropriate viewing window. Then visuall compare the heights of the water balloons. 3. Solve the Problem Enter the function f (t) = 6t + 8t + 5 into our calculator and graph it. Compare the graphs to determine which function has a greater maimum value. 5 st water balloon nd water balloon 5 6 4 6 t You can see that the second water balloon reaches a height of about 5 feet, while the first water balloon reaches a height of onl about feet. So, the second water balloon went higher. 4. Look Back Use the maimum feature to determine that the maimum value of f (t) = 6t + 8t + 5 is 5. Use a straightedge to represent a height of 5 feet on the graph that represents the second water balloon to clearl see that the second water balloon went higher. 5 st water balloon nd water balloon cm 3 4 5 6 7 8 9 3 4 5 5 Maimum X=.4999988 Y=5 6 4 6 t Monitoring Progress Help in English and Spanish at BigIdeasMath.com. Which balloon is in the air longer? Eplain our reasoning.. Which balloon reaches its maimum height faster? Eplain our reasoning. Section 8.3 Graphing f() = a + b + c 435
8.3 Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. VOCABULARY Eplain how ou can tell whether a quadratic function has a maimum value or a minimum value without graphing the function.. DIFFERENT WORDS, SAME QUESTION Consider the quadratic function f () = + 8 + 4. Which is different? Find both answers. What is the maimum value of the function? What is the greatest number in the range of the function? What is the -coordinate of the verte of the graph of the function? What is the ais of smmetr of the graph of the function? Monitoring Progress and Modeling with Mathematics In Eercises 3 6, find the verte, the ais of smmetr, and the -intercept of the graph. 3. 4 4. 4 4 7. = 3 6 + 5 8. f () = 3 4 9. ERROR ANALYSIS Describe and correct the error in finding the ais of smmetr of the graph of = 3 +. = b a = (3) = The ais of smmetr is =. 5. 4 4 6. 4 In Eercises 7, find (a) the ais of smmetr and (b) the verte of the graph of the function. (See Eample.) 7. f () = 4 8. = 3 + 5. ERROR ANALYSIS Describe and correct the error in graphing the function f () = + 4 + 3. The ais of smmetr is = b a = 4 () =. f () = 6 + 4() + 3 = 5 So, the verte is (, 5). The -intercept is 3. So, the points (, 3) and (4, 3) lie on the graph. 4 9. = 9 8. f () = 6 + 4. f () = 5 4 + 4. = 4 + 9 8 In Eercises 3 8, graph the function. Describe the domain and range. (See Eample.) 3. f () = + + 4 4. = 4 + 4 + 3 5. = 8 6 9 6. f () = 5 + 7 3 In Eercises 6, tell whether the function has a minimum value or a maimum value. Then find the value. (See Eample 3.). = 3 8 + 5. f () = 5 + + 7 3. f () = 4 + 4 4. = + 3 5. = + 6 6. f () = 5 5 + 7 436 Chapter 8 Graphing Quadratic Functions
7. MODELING WITH MATHEMATICS The function shown represents the height h (in feet) of a firework t seconds after it is launched. The firework eplodes at its highest point. (See Eample 4.) 3. ATTENDING TO PRECISION The verte of a parabola is (3, ). One point on the parabola is (6, 8). Find another point on the parabola. Justif our answer. 3. MAKING AN ARGUMENT Your friend claims that it is possible to draw a parabola through an two points with different -coordinates. Is our friend correct? Eplain. h = 6t + 8t USING TOOLS In Eercises 33 36, use the minimum or maimum feature of a graphing calculator to approimate the verte of the graph of the function. 33. =.5 + 3 34. = 6. + 4.8 a. When does the firework eplode? 35. = π + 3 b. At what height does the firework eplode? 36. =.5 5/3 + 8. MODELING WITH MATHEMATICS The function h(t) = 6t + 6t represents the height (in feet) of a horse t seconds after it jumps during a steeplechase. a. When does the horse reach its maimum height? b. Can the horse clear a fence that is 3.5 feet tall? If so, b how much? 37. MODELING WITH MATHEMATICS The opening of one aircraft hangar is a parabolic arch that can be modeled b the equation =.6 +.5, where and are measured in feet. The opening of a second aircraft hangar is shown in the graph. (See Eample 5.) c. How long is the horse in the air? 9. MODELING WITH MATHEMATICS The cable between 5 two towers of a suspension bridge can be modeled b the function shown, where and are measured in feet. The cable is at road level midwa between the towers. 5 5 a. Which aircraft hangar is taller? b. Which aircraft hangar is wider? = + 5 4 38. MODELING WITH MATHEMATICS An office a. How far from each tower shown is the lowest point of the cable? b. How high is the road above the water? c. Describe the domain and range of the function shown. 3. REASONING Find the ais of smmetr of the graph of the equation = a + b + c when b =. Can ou find the ais of smmetr when a =? Eplain. suppl store sells about 8 graphing calculators per month for $ each. For each $6 decrease in price, the store epects to sell eight more calculators. The revenue from calculator sales is given b the function R(n) = (unit price)(units sold), or R(n) = ( 6n)(8 + 8n), where n is the number of $6 price decreases. a. How much should the store charge to maimize monthl revenue? b. Using a different revenue model, the store epects to sell five more calculators for each $4 decrease in price. Which revenue model results in a greater maimum monthl revenue? Eplain. Section 8.3 hsnb_alg_pe_83.indd 437 Graphing f() = a + b + c 437 /5/5 8:4 AM
MATHEMATICAL CONNECTIONS In Eercises 39 and 4, (a) find the value of that maimizes the area of the figure and (b) find the maimum area. 46. CRITICAL THINKING Parabolas A and B contain the points shown. Identif characteristics of each parabola, if possible. Eplain our reasoning. 39. Parabola A Parabola B in. in. 6 in. 3 4 4..5 in. ( 4) ft 6 4 3 4 5 4 ft ( + ) ft 4. WRITING Compare the graph of g() = + 4 + with the graph of h() = 4 +. 4. HOW DO YOU SEE IT? During an archer competition, an archer shoots an arrow. The arrow follows the parabolic path shown, where and are measured in meters. 4 6 8 47. MODELING WITH MATHEMATICS At a basketball game, an air cannon launches T-shirts into the crowd. The function = 8 + 4 represents the path of a T-shirt. The function 3 = 4 represents the height of the bleachers. In both functions, represents vertical height (in feet) and represents horizontal distance (in feet). At what height does the T-shirt land in the bleachers? 48. THOUGHT PROVOKING One of two classic tangent line problems in calculus is finding the slope of a tangent line to a curve. An eample of a tangent line, which just touches the parabola at one point, is shown. 4 Approimate the slope of the tangent line to the graph of = at the point (, ). Eplain our reasoning. a. What is the initial height of the arrow? b. Estimate the maimum height of the arrow. c. How far does the arrow travel? 43. USING TOOLS The graph of a quadratic function passes through (3, ), (4, 7), and (9, ). Does the graph open up or down? Eplain our reasoning. 44. REASONING For a quadratic function f, what does f ( a) b represent? Eplain our reasoning. 49. PROBLEM SOLVING The owners of a dog shelter want to enclose a rectangular pla area on the side of their building. The have k feet of fencing. What is the maimum area of the outside enclosure in terms of k? (Hint: Find the -coordinate of the verte of the graph of the area function.) w pla area dog shelter building w 45. PROBLEM SOLVING Write a function of the form = a + b whose graph contains the points (, 6) and (3, 6). Maintaining Mathematical Proficienc Describe the transformation(s) from the graph of f () = to the graph of the given function. (Section 3.7) 5. q() = + 6 5. h() =.5 5. g() = + 5 53. p() = 3 + 438 Chapter 8 Graphing Quadratic Functions Reviewing what ou learned in previous grades and lessons