Using Intercept Form


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1 8.5 Using Intercept Form Essential Question What are some of the characteristics of the graph of f () = a( p)( q)? Using Zeros to Write Functions Work with a partner. Each graph represents a function of the form f () = ( p)( q) or f () = ( p)( q). Write the function represented b each graph. Eplain our reasoning. a. b. c. d. e. f. g. h. CONSTRUCTING VIABLE ARGUMENTS To be proficient in math, ou need to justif our conclusions and communicate them to others. Communicate Your Answer. What are some of the characteristics of the graph of f () = a( p)( q)? 3. Consider the graph of f () = a( p)( q). a. Does changing the sign of a change the intercepts? Does changing the sign of a change the intercept? Eplain our reasoning. b. Does changing the value of p change the intercepts? Does changing the value of p change the intercept? Eplain our reasoning. Section 8.5 Using Intercept Form 9
2 8.5 Lesson What You Will Learn Core Vocabular intercept form, p. 50 Graph quadratic functions of the form f () = a( p)( q). Use intercept form to find zeros of functions. Use characteristics to graph and write quadratic functions. Use characteristics to graph and write cubic functions. Graphing f ( ) = a( p)( q) You have alread graphed quadratic functions written in several different forms, such as f () = a + b + c (standard form) and g() = a( h) + k (verte form). Quadratic functions can also be written in intercept form, f () = a( p)( q), where a 0. In this form, the polnomial that defines a function is in factored form and the intercepts of the graph can be easil determined. Core Concept Graphing f ( ) = a( p)( q) The intercepts are p and q. The ais of smmetr is halfwa between (p, 0) and (q, 0). So, the ais of smmetr is = p + q. The graph opens up when a > 0, and the graph opens down when a < 0. (p, 0) = p + q (q, 0) Graphing f () = a( p)( q) Graph f () = ( + 1)( 5). Describe the domain and range. Step 1 Identif the intercepts. Because the intercepts are p = 1 and q = 5, plot ( 1, 0) and (5, 0). Step Find and graph the ais of smmetr. f() = ( + 1)( 5) = p + q = = 10 (, 9) Step 3 Find and plot the verte. 8 The coordinate of the verte is. To find the coordinate of the verte, substitute for and simplif. f () = ( + 1)( 5) = 9 So, the verte is (, 9). ( 1, 0) (5, 0) Step Draw a parabola through the verte and the points where the intercepts occur. The domain is all real numbers. The range is Chapter 8 Graphing Quadratic Functions
3 Graphing a Quadratic Function Graph f () = 8. Describe the domain and range. Step 1 Rewrite the quadratic function in intercept form. f () = 8 Write the function. = ( ) Factor out common factor. = ( + )( ) Difference of two squares pattern Step Identif the intercepts. Because the intercepts are p = and q =, plot (, 0) and (, 0). Step 3 Find and graph the ais of smmetr. (, 0) (, 0) = p + q + = 0 Step Find and plot the verte. The coordinate of the verte is 0. The coordinate of the verte is f (0) = (0) 8 = 8. So, the verte is (0, 8). Step 5 Draw a parabola through the verte and the points where the intercepts occur. (0, 8) f() = 8 The domain is all real numbers. The range is 8. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Graph the quadratic function. Label the verte, ais of smmetr, and intercepts. Describe the domain and range of the function. REMEMBER Functions have zeros, and graphs have intercepts. 1. f () = ( + )( 3). g() = ( )( + 1) 3. h() = 3 Using Intercept Form to Find Zeros of Functions In Section 8., ou learned that a zero of a function is an value for which f () = 0. You can use the intercept form of a function to find the zeros of the function. 5 Check Finding Zeros of a Function Find the zeros of f () = ( 1)( + ). To find the zeros, determine the values for which f () is 0. f () = ( 1)( + ) Write the function. 0 = ( 1)( + ) Substitute 0 for f(). 1 = 0 or + = 0 ZeroProduct Propert = 1 or = Solve for. So, the zeros of the function are and 1. Section 8.5 Using Intercept Form 51
4 Core Concept Factors and Zeros For an factor n of a polnomial, n is a zero of the function defined b the polnomial. Finding Zeros of Functions Find the zeros of each function. a. f () = 10 1 b. h() = ( 1)( 1) LOOKING FOR STRUCTURE The function in Eample (b) is called a cubic function. You can etend the concept of intercept form to cubic functions. You will graph a cubic function in Eample 7. Write each function in intercept form to identif the zeros. a. f () = 10 1 Write the function. = ( ) Factor out common factor. = ( + 3)( + ) Factor the trinomial. So, the zeros of the function are 3 and. b. h() = ( 1)( 1) Write the function. = ( 1)( + )( ) Difference of two squares pattern So, the zeros of the function are, 1, and. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Find the zero(s) of the function.. f () = ( )( 1) 5. g() = h() = ( 1) Using Characteristics to Graph and Write Quadratic Functions Use zeros to graph h() = 3. Graphing a Quadratic Function Using Zeros The function is in standard form. You know that the parabola opens up (a > 0) and the intercept is 3. So, begin b plotting (0, 3). ATTENDING TO PRECISION To sketch a more precise graph, make a table of values and plot other points on the graph. Notice that the polnomial that defines the function is factorable. So, write the function in intercept form and identif the zeros. h() = 3 Write the function. = ( + 1)( 3) Factor the trinomial. The zeros of the function are 1 and 3. So, plot ( 1, 0) and (3, 0). Draw a parabola through the points. h() = 3 5 Chapter 8 Graphing Quadratic Functions
5 Writing Quadratic Functions STUDY TIP In part (a), man possible functions satisf the given condition. The value a can be an nonzero number. To allow easier calculations, let a = 1. B letting a =, the resulting function would be f () = Write a quadratic function in standard form whose graph satisfies the given condition(s). a. verte: ( 3, ) b. passes through ( 9, 0), (, 0), and (, 0) a. Because ou know the verte, use verte form to write a function. f () = a( h) + k Verte form = 1( + 3) + Substitute for a, h, and k. = Find the product ( + 3). = Combine like terms. b. The given points indicate that the intercepts are 9 and. So, use intercept form to write a function. f () = a( p)( q) Intercept form = a( + 9)( + ) Substitute for p and q. Use the other given point, (, 0), to find the value of a. 0 = a( + 9)( + ) Substitute for and 0 for f(). 0 = a(5)( ) Simplif. = a Solve for a. Use the value of a to write the function. f () = ( + 9)( + ) Substitute for a. = 3 Simplif. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Use zeros to graph the function. 7. f () = ( 1)( ) 8. g() = + 1 Write a quadratic function in standard form whose graph satisfies the given condition(s). 9. intercepts: 1 and verte: (8, 8) 11. passes through (0, 0), (10, 0), and (, 1) 1. passes through ( 5, 0), (, 0), and (3, 1) Using Characteristics to Graph and Write Cubic Functions In Eample, ou etended the concept of intercept form to cubic functions. f () = a( p)( q)( r), a 0 Intercept form of a cubic function The intercepts of the graph of f are p, q, and r. Section 8.5 Using Intercept Form 53
6 Use zeros to graph f () = 3. Graphing a Cubic Function Using Zeros Notice that the polnomial that defines the function is factorable. So, write the function in intercept form and identif the zeros. f () = 3 Write the function. = ( ) Factor out. = ( + )( ) Difference of two squares pattern The zeros of the function are, 0, and. So, plot (, 0), (0, 0), and (, 0). To help determine the shape of the graph, find points between the zeros f() 3 3 f() = 3 Plot ( 1, 3) and (1, 3). Draw a smooth curve through the points. Writing a Cubic Function The graph represents a cubic function. Write the function. (0, 0) 8 (3, 1) (, 0) (5, 0) From the graph, ou can see that the intercepts are 0,, and 5. Use intercept form to write a function. f () = a( p)( q)( r) Intercept form = a( 0)( )( 5) Substitute for p, q, and r. = a()( )( 5) Simplif. Use the other given point, (3, 1), to find the value of a. 1 = a(3)(3 )(3 5) Substitute 3 for and 1 for f(). = a Solve for a. Use the value of a to write the function. f () = ()( )( 5) Substitute for a. = Simplif. The function represented b the graph is f () = Monitoring Progress Help in English and Spanish at BigIdeasMath.com Use zeros to graph the function. 13. g() = ( 1)( 3)( + 3) 1. h() = The zeros of a cubic function are 3, 1, and 1. The graph of the function passes through the point (0, 3). Write the function. 5 Chapter 8 Graphing Quadratic Functions
7 8.5 Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. COMPLETE THE SENTENCE The values p and q are of the graph of the function f () = a( p)( q).. WRITING Eplain how to find the maimum value or minimum value of a quadratic function when the function is given in intercept form. Monitoring Progress and Modeling with Mathematics In Eercises 3, find the intercepts and ais of smmetr of the graph of the function. 3. = ( 1)( + 3) 1. = ( )( 5) 5. = g() = 8 7. f () = ( + 5)( ) 8. h() = ( 3)( 11) 9. = = In Eercises 31 3, match the function with its graph. 31. = ( + 5)( + 3) 3. = ( + 5)( 3) 33. = ( 5)( + 3) 3. = ( 5)( 3) 5. f () = 5( + 7)( 5). g() = ( + 8) 3 In Eercises 7 1, graph the quadratic function. Label the verte, ais of smmetr, and intercepts. Describe the domain and range of the function. (See Eample 1.) 7. f () = ( + )( + 1) 8. = ( )( + ) 35. = ( + 5)( 5) 3. = ( + 3)( 3) A. 8 B = ( + )( ) 10. h() = ( 7)( 3) g() = 5( + 1)( + ) 1. = ( 3)( + ) In Eercises 13 0, graph the quadratic function. Label the verte, ais of smmetr, and intercepts. Describe the domain and range of the function. (See Eample.) 13. = 9 1. f () = 8 C. 8 D h() = = q() = p() = + 7 E. F. 19. = = In Eercises 1 30, find the zero(s) of the function. (See Eamples 3 and.) 1. = ( )( 10). f () = 1 ( + 5)( 1) g() = + 5. = Section 8.5 Using Intercept Form 55
8 In Eercises 37, use zeros to graph the function (See Eample 5.) Dnamic Solutions available (, at 3) BigIdeasMath.com (, 0) (10, 0) 37. f () = ( + )( ) 38. g() = 3( + 1)( + 7) 39. = = = h() = (, 0) (, 0) 8 (8, ) ERROR ANALYSIS In Eercises 3 and, describe and correct the error in finding the zeros of the function. 3.. = 5( + 3)( ) The zeros of the function are 3 and. = ( + )( 9) The zeros of the function are and 9. In Eercises 5 5, write a quadratic function in standard form whose graph satisfies the given condition(s). (See Eample.) 5. verte: (7, 3). verte: (, 8) 7. intercepts: 1 and intercepts: and 5 9. passes through (, 0), (3, 0), and (, 18) 50. passes through ( 5, 0), ( 1, 0), and (, 3) 51. passes through (7, 0) 5. passes through (0, 0) and (, 0) 53. ais of smmetr: = 5 5. increases as increases when < ; decreases as increases when >. 55. range: 3 5. range: 10 In Eercises 57 0, write the quadratic function represented b the graph ( 3, 0) 8 (, 0) (1, 8) (1, 0) (7, 0) (, 5) In Eercises 1 8, use zeros to graph the function. (See Eample 7.) 1. = 5( + )( ). f () = ( + 9)( + 3) 3. h() = ( )( + )( + 7). = ( + 1)( 5)( ) 5. f () = = = g() = In Eercises 9 7, write the cubic function represented b the graph. (See Eample 8.) ( 1, 0) (0, 0) (1, ) (, 0) (, 0) 0 ( 7, 0) (0, 0) (, 0) ( 3, 0) ( 1, 3) (0, 0) (, 0) (1, 0) (, 0) (3, 0) (5, 0) In Eercises 73 7, write a cubic function whose graph satisfies the given condition(s) intercepts:, 3, and intercepts: 7, 5, and passes through (1, 0) and (7, 0) 7. passes through (0, ) 5 Chapter 8 Graphing Quadratic Functions
9 In Eercises 77 80, all the zeros of a function are given. Use the zeros and the other point given to write a quadratic or cubic function represented b the table. 8. MODELING WITH MATHEMATICS A professional basketball plaer s shot is modeled b the function shown, where and are measured in feet = ( ) (3, 0) a. Does the plaer make the shot? Eplain. b. The basketball plaer releases another shot from the point (13, 0) and makes the shot. The shot also passes through the point (10, 1.). Write a quadratic function in standard form that models the path of the shot. In Eercises 81 8, sketch a parabola that satisfies the given conditions intercepts: and ; range: 3 8. ais of smmetr: = ; passes through (, 15) 83. range: 5; passes through (0, ) 8. intercept: ; intercept: 1; range: USING STRUCTURE In Eercises 87 90, match the function with its graph. 87. = = = = MODELING WITH MATHEMATICS Satellite dishes are shaped like parabolas to optimall receive signals. The cross section of a satellite dish can be modeled b the function shown, where and are measured in feet. The ais represents the top of the opening of the dish. 1 = ( ) 8 A. B. C. D. 1 a. How wide is the satellite dish? b. How deep is the satellite dish? c. Write a quadratic function in standard form that models the cross section of a satellite dish that is feet wide and 1.5 feet deep. 91. CRITICAL THINKING Write a quadratic function represented b the table, if possible. If not, eplain wh Section 8.5 Using Intercept Form 57
10 9. HOW DO YOU SEE IT? The graph shows the parabolic arch that supports the roof of a convention center, where and are measured in feet a. The arch can be represented b a function of the form f () = a( p)( q). Estimate the values of p and q. b. Estimate the width and height of the arch. Eplain how ou can use our height estimate to calculate a. ANALYZING EQUATIONS In Eercises 93 and 9, (a) rewrite the quadratic function in intercept form and (b) graph the function using an method. Eplain the method ou used. 93. f () = 3( + 1) g() = ( 1) 95. WRITING Can a quadratic function with eactl one real zero be written in intercept form? Eplain. 9. MAKING AN ARGUMENT Your friend claims that an quadratic function can be written in standard form and in verte form. Is our friend correct? Eplain. 97. PROBLEM SOLVING Write the function represented b the graph in intercept form. 80 (1, 0) (, 0) ( 5, 0) (8, 0) 10 (, 0) 10 (0, 3) 98. THOUGHT PROVOKING Sketch the graph of each function. Eplain our procedure. a. f () = ( 1)( ) b. g() = ( 1)( ) 99. REASONING Let k be a constant. Find the zeros of the function f () = k k k 3 in terms of k. PROBLEM SOLVING In Eercises 100 and 101, write a sstem of two quadratic equations whose graphs intersect at the given points. Eplain our reasoning (, 0) and (, 0) 101. (3, ) and (7, ) Maintaining Mathematical Proficienc The scatter plot shows the amounts (in grams) of fat and the numbers of calories in 1 burgers at a fastfood restaurant. (Section.) 10. How man calories are in the burger that contains 1 grams of fat? 103. How man grams of fat are in the burger that contains 00 calories? 10. What tends to happen to the number of calories as the number of grams of fat increases? Reviewing what ou learned in previous grades and lessons Calories Burgers Determine whether the sequence is arithmetic, geometric, or neither. Eplain our reasoning. (Section.) , 11, 1, 33, 7, ,, 18, 5, , 18, 10,,, , 5, 9, 1, 3, Fat (grams) 58 Chapter 8 Graphing Quadratic Functions
4 B. 4 D. 4 F. 3. How can you use the graph of a quadratic equation to determine the number of real solutions of the equation?
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