6.3 Interpreting Vertex Form and Standard Form


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1 Name Class Date 6.3 Interpreting Verte Form and Standard Form Essential Question: How can ou change the verte form of a quadratic function to standard form? Resource Locker Eplore Identifing Quadratic Functions from Their Graphs Determine whether a function is a quadratic function b looking at its graph. If the graph of a function is a parabola, then the function is a quadratic function. If the graph of a function is not a parabola, then the function is not a quadratic function. Use a graphing calculator to graph each of the functions. Set the viewing window to show 10 to 10 on both aes. Determine whether each function is a quadratic function. A B Use a graphing calculator to graph ƒ () = + 1. Determine whether the function ƒ () = + 1 is a quadratic function. The function ƒ () = + 1 a quadratic function. C Use a graphing calculator to graph ƒ () = Houghton Mifflin Harcourt Publishing Compan D Determine whether the function ƒ () = is a quadratic function. The function ƒ () = E Use a graphing calculator to graph ƒ () = 2. a quadratic function. F Determine whether the function ƒ () = 2 is a quadratic function. The function ƒ () = 2 a quadratic function. Module Lesson 3
2 Use a graphing calculator to graph ƒ () = Determine whether the function ƒ () = is a quadratic function. The function ƒ () = a quadratic function. Use a graphing calculator to graph ƒ () =  (  3) Determine whether the function ƒ () =  (  3) is a quadratic function. The function ƒ () =  (  3) a quadratic function. Use a graphing calculator to graph ƒ () =. Determine whether the function ƒ () = is a quadratic function. The function ƒ () = a quadratic function. Reflect 1. How can ou determine whether a function is quadratic or not b looking at its graph? 2. Discussion How can ou tell if a function is a quadratic function b looking at the equation? Houghton Mifflin Harcourt Publishing Compan Module Lesson 3
3 Eplain 1 Identifing Quadratic Functions in Standard Form If a function is quadratic, it can be represented b an equation of the form = a 2 + b + c, where a, b, and c are real numbers and a 0. This is called the standard form of a quadratic equation. The ais of smmetr for a quadratic equation in standard form is given b the equation = _ b. The verte of a 2a quadratic equation in standard form is given b the coordinates (  _ b 2a, ƒ (  _ 2a) b ). Eample 1 A = Determine whether the function represented b each equation is quadratic. If so, give the ais of smmetr and the coordinates of the verte. = Compare to = a 2 + b + c. This is not a quadratic function because a = 0. B = 4 Rewrite the function in the form = a 2 + b + c. = Compare to = a 2 + b + c. This a quadratic function. If = 4 is a quadratic function, give the ais of smmetr. If = 4 is a quadratic function, give the coordinates of the verte. Reflect 3. Eplain wh the function represented b the equation = a 2 + b + c is quadratic onl when a 0. Houghton Mifflin Harcourt Publishing Compan 4. Wh might it be easier to determine whether a function is quadratic when it is epressed in function notation? 5. How is the ais of smmetr related to standard form? Your Turn Determine whether the function represented b each equation is quadratic = = Module Lesson 3
4
5 Eplain 2 Changing from Verte Form to Standard Form It is possible to write quadratic equations in various forms. Eample 2 Rewrite a quadratic function from verte form, = a (  h) 2 + k, to standard form, = a 2 + b + c. A = 4 (  6) = 4 ( ) + 3 Epand (  6) 2. = = Multipl. Simplif. The standard form of = 4 (  6) is = B = 3 ( + 2) 21 = 3 ( )  1 Epand ( + 2) 2. =  1 Multipl. = Simplif. The standard form of = 3 ( + 2) 21 is =. Reflect 8. If in = a (  h) 2 + k, a = 1, what is the simplified form of the standard form, = a 2 + b + c? Your Turn Rewrite a quadratic function from verte form, = a (  h) 2 + k, to standard form, = a 2 + b + c. 9. = 2 ( + 5) = 3 (  7) Houghton Mifflin Harcourt Publishing Compan Module Lesson 3
6 Eplain 3 Writing a Quadratic Function Given a Table of Values You can write a quadratic function from a table of values. Eample 3 Use each table to write a quadratic function in verte form, = a (  h ) 2 + k. Then rewrite the function in standard form, = a 2 + b + c. A The minimum value of the function occurs at = 3. The verte of the parabola is (3, 0). Substitute the values for h and k into = a (  h) 2 + k. = a (  (3)) 2 + 0, or = a ( + 3) 2 Use an point from the table to find a. = a ( + 3) 2 1 = a (2 + 3) 2 = a The verte form of the function is = 1 (  (3)) or = ( + 3) 2. Rewrite the function = ( + 3) 2 in standard form, = a 2 + b + c. = ( + 3) 2 = The standard form of the function is = B The minimum value of the function occurs at = 2. The verte of the parabola is is (2, 3). Substitute the values for h and k into = a ( h) 2 + k. = Use an point from the table to find a. a = Houghton Mifflin Harcourt Publishing Compan The verte form of the function is =. Rewrite the resulting function in standard form, = a 2 + b + c. = Reflect 11. How man points are needed to find an equation of a quadratic function? Module Lesson 3
7 Your Turn Use each table to write a quadratic function in verte form, = a (  h) 2 + k. Then rewrite the function in standard form, = a 2 + b + c. 12. The verte of the function is (2, 5) The verte of the function is (2, 7) P Eplain 4 Writing a Quadratic Function Given a Graph The graph of a parabola can be used to determine the corresponding function. Eample 4 Use each graph to find an equation for ƒ (t). A A house painter standing on a ladder drops a paintbrush, which falls to the ground. The paintbrush s height above the ground (in feet) is given b a function of the form ƒ (t) = a (t  h) 2 where t is the time (in seconds) after the paintbrush is dropped. The verte of the parabola is (h, k) = (0, 25). ƒ (t) = a (  h) 2 + k ƒ (t) = a (t  0) ƒ (t) = at Use the point (1, 9) to find a. Height (feet) (0, 25) (1, 9) Time (seconds) Houghton Mifflin Harcourt Publishing Compan ƒ (t) = at = a (1) = a The equation for the function is ƒ (t) = 16 t Module Lesson 3
8 B A rock is knocked off a cliff into the water far below. The falling rock s height above the water (in feet) is given b a function of the form ƒ (t) = a (t  h) 2 + k where t is the time (in seconds) after the rock begins to fall. Height (feet) (0, 40) (1, 24) Time (seconds) The verte of the parabola is (h, k) =. ƒ (t) = a (t  h) 2 + k 2 ƒ (t) = a ( ) t  +. ƒ (t) = Use the point to find a. Images/Corbis ƒ (t) = a t 2 + = a a = 2 + The equation for the function is ƒ (t) =. Reflect 14. Identif the domain and eplain wh it makes sense for this problem. 15. Identif the range and eplain wh it makes sense for this problem. Module Lesson 3
9 Your Turn 16. The graph of a function in the form ƒ () = a (  h) 2 + k, is shown. Use the graph to find an equation for ƒ () (1, 1) (3, 3) A roofer accidentall drops a nail, which falls to the ground. The nail s height above the ground (in feet) is given b a function of the form ƒ (t) = a (t  h) 2 + k, where t is the time (in seconds) after the nail drops. Use the graph to find an equation for ƒ (t). Height (feet) (0, 45) (1, 29) Time (seconds) t Module Lesson 3
10 Elaborate 18. Describe the graph of a quadratic function. 19. What is the standard form of the quadratic function? 20. Can an quadratic function in verte form be written in standard form? 21. How man points are needed to write a quadratic function in verte form, given the table of values? 22. If a graph of the quadratic function is given, how do ou find the verte? 23. Essential Question CheckIn What can ou do to change the verte form of a quadratic function to standard form? Evaluate: Homework and Practice Determine whether each function is a quadratic function b graphing. 1. ƒ () = ƒ () = 1 _ ƒ () = ƒ () = 23 Houghton Mifflin Harcourt Publishing Compan Determine whether the function represented b each equation is quadratic. 5. = = = = Which of the following functions is a quadratic function? Select all that appl. a. 2 = + 3 d = 0 b = 31 e.  = 4 c. 5 = For ƒ () = , give the ais of smmetr and the coordinates of the verte. = b = (8) = 8 = 4 2a 2(1) 2 = (4) + 8(4)  14 = 30 Verte (4,30) Ais of Sm: = 4 Module Lesson 3
11 11. Describe the ais of smmetr of the graph of the quadratic function represented b the equation = a 2 + b + c. when b = 0. Rewrite each quadratic function from verte form, = a (  h) 2 + k, to standard form, = a 2 + b + c. 12. = 5 (  2) = 2 ( + 4) = 3 ( + 1) = 4 (  3) Eplain the Error Tim wrote = 6 ( + 2) 210 in standard form as = Find his error. 17. How do ou change from verte form, ƒ () = a (  h) 2 + k, to standard form, = a 2 + b + c? Houghton Mifflin Harcourt Publishing Compan Module Lesson 3
12 Use each table to write a quadratic function in verte form, = a (  h) 2 + k. Then rewrite the function in standard form, = a 2 + b + c. 18. The verte of the function is (6, 8) The verte of the function is (4, 7) The verte of the function is (2, 12) The verte of the function is (3, 10) Houghton Mifflin Harcourt Publishing Compan Module Lesson 3
13 H.O.T. Focus on Higher Order Thinking 22. Make a Prediction A ball was thrown off a bridge. The table relates the height of the ball above the ground in feet to the time in seconds after it was thrown. Use the data to write a quadratic model in verte form and convert it to standard form. Use the model to find the height of the ball at 1.5 seconds. Time (seconds) Height (feet) Multiple Representations A performer slips and falls into a safet net below. The function ƒ (t) = a (t  h) 2 + k, where t represents time (in seconds), gives the performer s height above the ground (in feet) as he falls. Use the graph to find an equation for ƒ (t). Height (feet) (0, 20) (1, 4) Time (seconds) Module Lesson 3
14 24. Represent RealWorld Problems After a heav snowfall, Ken and Karin made an igloo. The dome of the igloo is in the shape of a parabola, and the height of the igloo in inches is given b the function ƒ () = a (  h) 2 + k. Use the graph to find an equation for ƒ (). 48 (40, 48) Height (feet) (60, 36) Width (feet) Houghton Mifflin Harcourt Publishing Compan 25. Check for Reasonableness Tim hits a softball. The function ƒ (t) = a (t  h) 2 + k describes the height (in feet) of the softball, and t is the time (in seconds). Use the graph to find an equation for ƒ (t). Estimate how much time elapses before the ball hits the ground. Use the equation for the function and our estimate to eplain whether the equation is reasonable. Height (feet) (1.5, 40) (2, 36) Time (seconds) Module Lesson 3
15 Lesson Performance Task The table gives the height of a tennis ball t seconds after it has been hit, where the maimum height is 4 feet. Time (s) Height (ft) a. Use the data in the table to write the quadratic function ƒ (t) in verte form, where t is the time in seconds and ƒ (t) is the height of the tennis ball in feet. b. Rewrite the function found in part a in standard form. c. At what height was the ball originall hit? Eplain. Houghton Mifflin Harcourt Publishing Compan Module Lesson 3
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