Chapter Maintaining Mathematical Proficienc Find the -intercept of the graph of the linear equation. 1. = + 3. = 3 + 5 3. = 10 75. = ( 9) 5. 7( 10) = +. 5 + 15 = 0 Find the distance between the two points. 7. ( 1, 3 ), (, ). ( 5, 0 ), ( 9, ) 9. ( 3, 7 ), ( 10, ) 10. (, ), ( 3, 0) 11. ( 9, 1 ), ( 9, ) 1. ( 0, 5 ), (, ) 13. A student uses the Distance Formula to find the distance between two points ( a, b) and ( c, d ). What does the step ( c a) ( 0) + tell the student about the relationship between the two points? How could the student have found the distance between the two points using another method? Copright Big Ideas Learning, LLC Algebra 3
Name Date.1 Transformations of Quadratic Functions For use with Eploration.1 Essential Question How do the constants a, h, and k affect the graph of the quadratic function g( ) = a( h) + k? 1 EXPLORATION: Identifing Graphs of Quadratic Functions Work with a partner. Match each quadratic function with its graph. Eplain our reasoning. Then use a graphing calculator to verif that our answer is correct. a. g( ) = ( ) b. g( ) = ( ) + c. g( ) ( ) = + d. g( ) = 0.5( ) e. g( ) = ( ) f. g( ) ( ) = + + A. B. C. D. E. F. Algebra Copright Big Ideas Learning, LLC
.1 Transformations of Quadratic Functions (continued) Communicate Your Answer. How do the constants a, h, and k affect the graph of the quadratic function g( ) = a( h) + k? 3. Write the equation of the quadratic function whose graph is shown. Eplain our reasoning. Then use a graphing calculator to verif that our equation is correct. Copright Big Ideas Learning, LLC Algebra 5
Name Date.1 Notetaking with Vocabular For use after Lesson.1 In our own words, write the meaning of each vocabular term. quadratic function parabola verte of a parabola verte form Core Concepts Horizontal Translations ( ) f = ( ) = ( ) f h h Vertical Translations ( ) ( ) f = f + k = + k = ( h), h < 0 = = + k, k > 0 = = ( h), h > 0 = + k, k < 0 shifts left when h < 0 shifts right when h > 0 shifts down when k < 0 shifts up when k > 0 Notes: Algebra Copright Big Ideas Learning, LLC
.1 Notetaking with Vocabular (continued) Reflections in the -Ais ( ) f = ( ) ( ) f = = = Reflections in the -Ais ( ) f = ( ) ( ) f = = = = flips over the -ais Horizontal Stretches and Shrinks ( ) f ( ) = ( a) f a = = is its own reflection in the -ais. Vertical Stretches and Shrinks f = ( ) ( ) a f = a = (a), a > 1 = = a, a > 1 = = (a), 0 < a < 1 = a, 0 < a < 1 horizontal stretch (awa from -ais) when 0 < a < 1 horizontal shrink (toward -ais) when a > 1 vertical stretch (awa from -ais) when a > 1 vertical shrink (toward -ais) when 0 < a < 1 Notes: Copright Big Ideas Learning, LLC Algebra 7
Name Date.1 Notetaking with Vocabular (continued) Etra Practice f In Eercises 1, describe the transformation of ( ) Then graph the function. = represented b g. 1. g ( ) = +. g ( ) = ( 1) 3 3. g ( ) = ( + 9) 1 1 1 1. g ( ) = 7 5. g ( ) =. ( ) ( ) 1 3 g = 1 1 1 1 f 7. Consider the function ( ) ( ) the parent quadratic function. Then identif the verte. = 10 5 + 7. Describe the transformation of the graph of Algebra Copright Big Ideas Learning, LLC
. Characteristics of Quadratic Functions For use with Eploration. Essential Question What tpe of smmetr does the graph of f( ) = a( h) + k have and how can ou describe this smmetr? 1 EXPLORATION: Parabolas and Smmetr Work with a partner. a. Complete the table. Then use the values in the table to sketch the graph of the function f( ) = 1 on graph paper. 1 0 1 f() 3 5 f() b. Use the results in part (a) to identif the verte of the parabola. c. Find a vertical line on our graph paper so that when ou fold the paper, the left portion of the graph coincides with the right portion of the graph. What is the equation of this line? How does it relate to the verte? f d. Show that the verte form ( ) ( ) given in part (a). = 1 is equivalent to the function Copright Big Ideas Learning, LLC Algebra 9
Name Date. Characteristics of Quadratic Functions (continued) EXPLORATION: Parabolas and Smmetr Work with a partner. Repeat Eploration 1 for the function given b f( ) = 1 + + 3 = 1( 3) +. 3 3 1 0 1 f() 3 5 f() Communicate Your Answer 3. What tpe of smmetr does the graph of the parabola f ( ) = a( h) + k have and how can ou describe this smmetr?. Describe the smmetr of each graph. Then use a graphing calculator to verif our answer. a. f( ) = ( 1) + b. f( ) = ( + 1) c. f( ) ( ) = 3 + 1 1 d. f( ) = ( + ) e. f( ) = + 3 f. ( ) ( ) f = 3 5 + 30 Algebra Copright Big Ideas Learning, LLC
. Notetaking with Vocabular For use after Lesson. In our own words, write the meaning of each vocabular term. ais of smmetr standard form minimum value maimum value intercept form Core Concepts Properties of the graph of f( ) = a + b + c = a + b + c, a > 0 (0, c) = a + b + c, a < 0 = b a b = a (0, c) The parabola opens up when a > 0 and open down when a < 0. The graph is narrower than the graph of f( ) = when a > 1 and wider when a < 1. The ais of smmetr is b b b = and the verte is, f. a a a The -intercept is c. So, the point ( 0, c ) is on the parabola. Notes: Copright Big Ideas Learning, LLC Algebra 31
Name Date. Notetaking with Vocabular (continued) Minimum and Maimum Values For the quadratic function f ( ) = a + b + c, the -coordinate of the verte is the minimum value of the function when a > 0 and the maimum value when a < 0. decreasing a > 0 a < 0 minimum = b a increasing increasing = b a maimum decreasing b Minimum value: f a Domain: All real numbers b Range: f a Decreasing to the left of Increasing to the right of b = a b = a b Maimum value: f a Domain: All real numbers b Range: f a Increasing to the left of Decreasing to the right of b = a b = a Notes: Properties of the graph of f( ) = a( p)( q) Because f( p) 0 and f( q) 0, = = p and q are the -intercepts of the graph of the function. The ais of smmetr is halfwa between ( p, 0 ) and ( q, 0 ). p + q So, the ais of smmetr is =. The parabola opens up when a > 0 and opens down when a < 0. (p, 0) = p + q = a( p)( q) (q, 0) Notes: 3 Algebra Copright Big Ideas Learning, LLC
. Notetaking with Vocabular (continued) Etra Practice In Eercises 1 3, graph the function. Label the verte and ais of smmetr. Find the minimum or maimum value of the function. Describe the domain and range of the function, and where the function is increasing and decreasing. t = 3 3 1 1. f( ) = ( + 1). ( ) = 5 3. ( ) In Eercises and 5, graph the function. Label the -intercept(s), verte, and ais of smmetr.. f( ) = ( + )( 3) 5. f( ) = 7( ) f = 0.0005 + 0.17 + 3, where is the distance from home plate (in feet) and is the height of the ball above the ground (in feet). What is the highest point this ball will reach? If the ball was hit to center field which has an foot fence located 10 feet from home plate, was this hit a home run? Eplain.. A softball plaer hits a ball whose path is modeled b ( ) Copright Big Ideas Learning, LLC Algebra 33
Name Date.3 Focus of a Parabola For use with Eploration.3 Essential Question What is the focus of a parabola? 1 EXPLORATION: Analzing Satellite Dishes Go to BigIdeasMath.com for an interactive tool to investigate this eploration. Work with a partner. Vertical ras enter a satellite dish whose cross section is a parabola. When the ras hit the parabola, the reflect at the same angle at which the entered. (See Ra 1 in the figure.) a. Draw the reflected ras so that the intersect the -ais. b. What do the reflected ras have in common? c. The optimal location for the receiver of the satellite dish is at a point called the focus of the parabola. Determine the location of the focus. Eplain wh this makes sense in this situation. Ra Ra Ra incoming angle outgoing angle 1 1 = 1 1 3 Algebra Copright Big Ideas Learning, LLC
.3 Focus of a Parabola (continued) EXPLORATION: Analzing Spotlights Go to BigIdeasMath.com for an interactive tool to investigate this eploration. Work with a partner. Beams of light are coming from the bulb in a spotlight, located at the focus of the parabola. When the beams hit the parabola, the reflect at the same angle at which the hit. (See Beam 1 in the figure.) Draw the reflected beams. What do the have in common? Would ou consider this to be the optimal result? Eplain. outgoing angle 1 = Beam 1 bulb incoming angle Beam 1 Beam 1 Communicate Your Answer 3. What is the focus of a parabola?. Describe some of the properties of the focus of a parabola. Copright Big Ideas Learning, LLC Algebra 35
Name Date.3 Notetaking with Vocabular For use after Lesson.3 In our own words, write the meaning of each vocabular term. focus directri Core Concepts Standard Equations of a Parabola with Verte at the Origin Vertical ais of smmetr ( = 0) 1 Equation: = p Focus: ( 0, p ) Directri: = p verte: (0, 0) focus: (0, p) directri: = p directri: = p p > 0 p < 0 verte: (0, 0) focus: (0, p) Horizontal ais of smmetr ( = 0) Equation: Focus: ( p, 0) Directri: 1 = p = p directri: = p focus: (p, 0) verte: (0, 0) focus: (p, 0) verte: (0, 0) p > 0 p < 0 directri: = p Notes: 3 Algebra Copright Big Ideas Learning, LLC
.3 Notetaking with Vocabular (continued) Standard Equations of a Parabola with Verte at ( h, k) Vertical ais of smmetr ( = h) = 1 h + k p Equation: ( ) = h (h, k + p) = h Focus: ( h, k + p) = k p Directri: = k p = k p (h, k) (h, k) (h, k + p) p > 0 p < 0 Horizontal ais of smmetr ( = k) = 1 k + h p Equation: ( ) Focus: ( h + p, k) Directri: = h p Notes: = k = h p (h, k) (h, k) (h + p, k) = k (h + p, k) = h p p > 0 p < 0 Etra Practice In Eercises 1 and, use the Distance Formula to write an equation of the parabola. 1. focus: ( 0, ) directri: =. verte: ( 0, 0 ) focus: ( 0, 1 ) Copright Big Ideas Learning, LLC Algebra 37
Name Date.3 Notetaking with Vocabular (continued) In Eercises 3 5, identif the focus, directri, and ais of smmetr of the parabola. Graph the equation. 3. =. 5 1 0 3 + = 5. ( ) = + 1 3 In Eercises, write an equation of the parabola shown.. 7. focus verte = 7 1 directri verte. = 3 5 directri verte 9. The cross section of a parabolic sound reflector at the Olmpics has a diameter of 0 inches and is 5 inches deep. Write an equation that represents the cross section of the reflector with its verte at ( 0, 0) and its focus to the left of the verte. 3 Algebra Copright Big Ideas Learning, LLC
. Modeling with Quadratic Functions For use with Eploration. Essential Question How can ou use a quadratic function to model a real-life situation? 1 EXPLORATION: Modeling with a Quadratic Function Work with a partner. The graph shows a quadratic function of the form P() t = at + bt + c which approimates the earl profits for a compan, where Pt () is the profit in ear t. a. Is the value of a positive, negative, or zero? Eplain. Yearl profit (dollars) P P(t) = at + bt + c b. Write an epression in terms of a and b that represents the ear t when the compan made the least profit. Year t c. The compan made the same earl profits in 00 and 01. Estimate the ear in which the compan made the least profit. d. Assume that the model is still valid toda. Are the earl profits currentl increasing, decreasing, or constant? Eplain. EXPLORATION: Modeling with a Graphing Calculator Go to BigIdeasMath.com for an interactive tool to investigate this eploration. Work with a partner. The table shows the heights h (in feet) of a wrench t seconds after it has been dropped from a building under construction. Time, t 0 1 3 Height, h 00 3 33 5 1 a. Use a graphing calculator to create a scatter plot of the data, as shown at the right. Eplain wh the data appear to fit a quadratic model. 00 0 0 5 Copright Big Ideas Learning, LLC Algebra 39
Name Date. Modeling with Quadratic Functions (continued) EXPLORATION: Modeling with a Graphing Calculator (continued) b. Use the quadratic regression feature to find a quadratic model for the data. c. Graph the quadratic function on the same screen as the scatter plot to verif that it fits the data. d. When does the wrench hit the ground? Eplain. Communicate Your Answer 3. How can ou use a quadratic function to model a real-life situation?. Use the Internet or some other reference to find eamples of real-life situations that can be modeled b quadratic functions. 0 Algebra Copright Big Ideas Learning, LLC
. Notetaking with Vocabular For use after Lesson. In our own words, write the meaning of each vocabular term. average rate of change sstem of three linear equations Core Concepts Writing Quadratic Equations Given a point and the verte ( h, k ) Use verte form: = a( h) + k Given a point and -intercepts p and q Use intercept form: = a ( p)( q) Given three points Write and solve a sstem of three equations in three variables. Notes: Copright Big Ideas Learning, LLC Algebra 1
Name Date. Notetaking with Vocabular (continued) Etra Practice In Eercises 1, write an equation of the parabola in verte form. 1.. (, ) (0, 3) (, 1) (, ) 3. passes through ( 3, 0) and has verte ( 1, ). passes through (, 7) and has verte (, 5) In Eercises 5, write an equation of the parabola in intercept form. 5. (1, 5). 10 1 1 ( 3, ) (, 0) (, 0) 10 (, 0) ( 3, 0) 1 7. -intercepts of 5 and ; passes through ( 1, ). -intercepts of 7 and 10; passes through (, 7) Algebra Copright Big Ideas Learning, LLC
. Notetaking with Vocabular (continued) In Eercises 9 11, analze the differences in the outputs to determine whether the data are linear, quadratic or neither. If linear or quadratic, write an equation that fits the data. 9. Time (seconds), 1 3 5 Distance (feet), 1 37 30 00 10. Time (das), 0 3 9 1 15 Height (inches), 3 30 1 1 11. Time (ears), 1 3 5 Profit (dollars), 5 15 5 135 05 115 1. The table shows a universit s budget (in millions of dollars) over a 10-ear period, where = 0 represents the first ear in the 10-ear period. Years, 0 1 3 5 7 9 Budget, 5 3 0 5 9 11 1 10 150 a. Use a graphing calculator to create a scatter plot. Which better represents the data, a line or a parabola? Eplain. b. Use the regression feature of our calculator to find the model that best fits the data. c. Use the model in part (b) to predict when the budget of the universit is $500,000,000.00. Copright Big Ideas Learning, LLC Algebra 3