Section 2.3 Quadratic Functions and Models

Transcription

1 Section.3 Quadratic Functions and Models Quadratic Function A function f is a quadratic function if f ( ) a b c Verte of a Parabola The verte of the graph of f( ) is V or b v a V or b y yv f a Verte Point b, f b a a f ( ) 4 b 4 a (1) b () y f f a 4() 6 Verte point:, 6 Ais of Symmetry: V b a Ais of Symmetry: = Minimum or Maimum Point If a > 0 f() has a minimum point If a < 0 f() has a maimum verte point V, V y Range If a > 0 V y, If a < 0, Vy Minimum 6,, 6 Domain:, f ( ) 4 y -intercept Symmetry Line Minimum / Verte point 30

2 Eample For the graph of the function a. Find the verte point 6 (1) 3 f ( ) 6 3 y f( 3) ( 3) 6( 3) 3 6 Verte point (3, 6) b. Find the line of symmetry: = 3 c. State whether there is a maimum or minimum value and find that value Minimum point, value (3, 6) d. Find the -intercept 6 6 4(1)(3) (1) e. Find the y-intercept y = 3 f. Find the range and the domain of the function. Range: [6, ) Domain: (, ) g. Graph the function and label, show part a thru d on the plot below Symmetry: = 3 f ( ) 6 3 y -intercept Verte Point / Min (3, 6) h. On what intervals is the function increasing? Decreasing? Decreasing: (,3) Increasing: (3, ) 31

3 Eample Find the ais and verte of the parabola having equation b a 4 () 1 Ais of the parabola: 1 f ( ) 4 5 y f( 1) ( 1) 4( 1) 5 3 Verte point: 1,3 Maimizing Area You have 10 ft of fencing to enclose a rectangular region. Find the dimensions of the rectangle that maimize the enclosed area. What is the maimum area? P l w 10 l w 60 l w l 60 w A lw (60 ww ) 60w w w 60w Verte: w ( 1) l 60 w 30 A lw (30)(30) 900 ft 3

4 Eample A stone mason has enough stones to enclose a rectangular patio with 60 ft of stone wall. If the house forms one side of the rectangle, what is the maimum area that the mason can enclose? What should the dimensions of the patio be in order to yield this area? P l w 60 l 60 w A lw (60 w)w 60ww w 60w w b a 60 ( ) 15 ft l 60 w 60 (15) 30 ft Area = (15)(30) = 450 ft Position Function (Projectile Motion) Eample A model rocket is launched with an initial velocity of 100 ft/sec from the top of a hill that is 0 ft high. Its height t seconds after it has been launched is given by the function s( t) 16t 100t 0. Determine the time at which the rocket reaches its maimum height and find the maimum height. t b a 100 ( 16) 3.15 sec st ( 3.15) 16( 3.15) 100( 3.15) ft 33

5 Eercises Section.3 Quadratic Functions and Models 1. Give the verte, ais, domain, and range. Then, graph the function. Give the verte, ais, domain, and range. Then, graph the function f ( ) 6 5 f ( ) 6 5 f 4 3. Give the verte, ais, domain, and range. Then, graph the function f Give the verte, ais, domain, and range. Then, graph the function 5. You have 600 ft. of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maimize the area. What is the largest area that can be enclosed? 6. A picture frame measures 8 cm by 3 cm and is of uniform width. What is the width of the frame if 19 cm of the picture shows? 7. An open bo is made from a 10-cm by 0-cm of tin by cutting a square from each corner and folding up the edges. The area of the resulting base is 96 cm. What is the length of the sides of the squares? 34

6 8. A fourth-grade class decides to enclose a rectangular garden, using the side of the school as one side of the rectangle. What is the maimum area that the class can enclose with 3 ft. of fence? What should the dimensions of the garden be in order to yield this area? 9. A rancher needs to enclose two adjacent rectangular corrals, one for cattle and one for sheep. If a river forms one side of the corrals and 40 yd of fencing is available, what is the largest total area that can be enclosed? 10. A Norman window is a rectangle with a semicircle on top. Sky Blue Windows is designing a Norman window that will require 4 ft of trim on the outer edges. What dimensions will allow the maimum amount of light to enter a house? 35

7 11. A frog leaps from a stump 3.5 ft. high and lands 3.5 ft. from the base of the stump. It is determined that the height of the frog as a function of its distance,, from the base of the stump is given by the function h where h is in feet. a) How high is the frog when its horizontal distance from the base of the stump is ft.? b) At what two distances from the base of the stump after is jumped was the frog 3.6 ft. above the ground? c) At what distance from the base did the frog reach its highest point? d) What was the maimum height reached by the frog? 36

8 Section.3 Quadratic Functions Eercise Give the verte, ais, domain, and range. Then, graph the function f ( ) 6 5 Verte: b a 6 (1) 3 y f( 3) ( 3) 6( 3) 5 4 Verte point: 3, 4 Ais of symmetry: 3 Domain:, Range: 4, y Eercise Give the verte, ais, domain, and range. Then, graph the function f ( ) 6 5 Verte: b a 6 ( 1) 3 y f( 3) ( 3) 6( 3) 5 4 Verte point: 3, 4 Ais of symmetry: 3 Domain:, Range:,4 y 40

9 Eercise Graph the quadratic. Give the verte, ais of symmetry, domain, and range: f 4 Verte point: b 4 a 1 f 4 The verte point:, Ais of symmetry is: Domain:, Range:, (Since function has a minimum) To graph: find another point: 0 y f 0 y Eercise Graph the quadratic. Give the verte, ais of symmetry, domain, and range: f 16 6 Verte point: b 16 4 a f The verte point: 4, 6 Ais of symmetry is: 4 Domain:, Range:,6 (Since function has a maimum) To graph: find another point: 3 y f 3 4 y 41

10 Eercise You have 600 ft of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maimize the area. What is the largest area that can be enclosed? P l w 600 l w l 600 w A lw (600 ww ) 600w w w 600w Verte: w ( ) l 600 w 300 A lw (300)(150) ft Eercise A picture frame measures 8 cm by 3 cm and is of uniform width. What is the width of the frame if 19 cm of the picture shows? Area of the picture = ( 3 )(8 ) ( 8)( )

11 Eercise An open bo is made from a 10-cm by 0-cm of tin by cutting a square from each corner and folding up the edges. The area of the resulting base is 96 cm. What is the length of the sides of the squares? Area of the base Solve for, 13 The length of the sides of the squares is 3-cm Eercise A fourth-grade class decides to enclose a rectangular garden, using the side of the school as one side of the rectangle. What is the maimum area that the class can enclose with 3 ft. of fence? What should the dimensions of the garden be in order to yield this area? Perimeter: P l w 3 Area: A lw A (3 w) w 3w w w 3w l 3 w Verte: w 3 8 ( ) 8 l 3 16 A lw 18 ft

12 Eercise A rancher needs to enclose two adjacent rectangular corrals, one for cattle and one for sheep. If a river forms one side of the corrals and 40 yd of fencing is available, what is the largest total area that can be enclosed? Perimeter: P l 3w 40 Area: A lw A (40 3 w) w 40w 3w 3w 40w l 40 3w Verte: w ( 3) l A lw yd Eercise A Norman window is a rectangle with a semicircle on top. Sky Blue Windows is designing a Norman window that will require 4 ft. of trim on the outer edges. What dimensions will allow the maimum amount of light to enter a house? Perimeter of the semi-circle 1 Perimeter of the rectangle y Total perimeter: y 4 y 4 y 1 1 y Area

13 4 b a 4 4 y Eercise A frog leaps from a stump 3.5 ft. high and lands 3.5 ft. from the base of the stump. It is determined that the height of the frog as a function of its distance,, from the base of the stump is given by the function h where h is in feet. a) How high is the frog when its horizontal distance from the base of the stump is ft.? b) At what two distances from the base of the stump after is jumped was the frog 3.6 ft. above the ground? c) At what distance from the base did the frog reach its highest point? d) What was the maimum height reached by the frog? a) At ft. Find h h ft h b) Solve for : 0.1, 1.4 ft c) The distance from the base for the frog to reach the highest point is b ft a d) Maimum height: h ft h

14 Eercise For the graph of the function a. Find the verte point 6 3 (1) f ( ) 6 3 y f( 3) ( 3) 6( 3) 3 6 Verte point (3, 6) b. Find the line of symmetry: = 3 c. State whether there is a maimum or minimum value and find that value Minimum point, value (3, 6) d. Find the zeros of f() 6 6 4(1)(3) (1) e. Find the y-intercept y 3 f. Find the range and the domain of the function. Range: [6, ) Domain: (, ) 3 6 g. Graph the function and label, show part a thru d on the plot below: Symmetry: = 3 y f ( ) 6 3 Zero s Verte Point / Min (3, 6) h. On what intervals is the function increasing? Decreasing? Decreasing: (, 3) Increasing: (3, ) 46