Section 6.4. Applications of Quadratic Functions

Section 6.4: Applications of Quadratic Functions Chapter 6: Quadratic Equations For example: A rocket is launches upward into the air, the equation ℎ(𝑡) = −16𝑡 2 + 320𝑡 gives the height (ℎ(𝑡)) in feet that the rocket is above the ground (𝑡) seconds after it is thrown. a) Make a table of values and sketch of graph of the function. b) What do the numbers on the x (horizontal axis) represent? c) What do the numbers on the y (vertical axis) represent? d) Identify the vertex. e) What do the coordinates of the vertex represent? f) When does the ball reach its maximum height? g) What is the maximum height of the ball? h) Find the 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡(𝑠) using Algebra. i) Give a physical interpretation of any 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 that is greater than or equal to zero. j) Find the practical domain of the function. k) Find the practical range of the function. a) 𝑡 8 9 10 11 12 ℎ(𝑡) 1536 1584 1600 1584 1536 −𝑏 Use 𝑡 = 2𝑎 for number in middle t-column 𝑎 = −16, 𝑏 = 320 −320 # middle of t-column 𝑡 = 2(−16) = 10 I used a calculator to get the y-coordinates. You may do the same. Continued on next page: Section 6.4: Applications of Quadratic Functions Chapter 6: Quadratic Equations b) 𝑥 / 𝑡 − 𝑎𝑥𝑖𝑠 represents the time in seconds that the rocket is traveling. c) 𝑦 / ℎ(𝑡) axis represents the height of the rocket in feet. d) Vertex (10, 1600) e) The 𝑥 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 (𝑡 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒) of the vertex represents the time of 10 seconds. The 𝑦 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 (ℎ(𝑡) − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒) of the vertex represents a height of 1600 ft. f) The vertex is a maximum point. The 𝑥 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 of the vertex gives the time the rocket reaches its maximum height. The ball reaches its maximum height at 10 seconds. h) The maximum height is the 𝑦 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 of the vertex. The maximum height is 1600 meters. i) To find the 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡(𝑡 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑠) set the function equal to zero and solve for t. −16𝑡 2 + 320𝑡 = 0 −16𝑡(𝑡 − 20) = 0 −16𝑡 = 0 𝑡 − 20 = 0 𝑡 = 0 𝑡 = 20 x (t)-intercepts: (0,0) (20,0) j) The 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑠 give the times where the rocket is on the ground as its height is 0 feet. The rocket is on the ground at the start of the launch. The rocket lands after 20 seconds. k) The practical domain is: Domain [0,20] Negative values indicate time before the rocket has launched. Values larger than 20 indicate time after the rocket has landed. l) The practical range is: Range [0,1600] The range represents the height of the rocket. It never gets lower than 0 feet nor higher than 1600 feet. Section 6.4: Applications of Quadratic Functions Chapter 6: Quadratic Equations For example: The depth (in feet) of the snow at the base of a ski run over a 1-year period is modeled by the function: ℎ(𝑡) = 4(𝑡 − 9)2 + 15, where 𝑡 = 1 represents January, 𝑡 = 2 represents February, and so on. a) Make a table of values and sketch of graph of the function. b) What do the numbers on the 𝑥 / 𝑡 (ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑎𝑥𝑖𝑠) represent? c) What do the numbers on the 𝑦 / ℎ(𝑡) (𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑎𝑥𝑖𝑠) represent? d) Identify the vertex. e) What do the coordinates of the vertex represent? f) How low did the water level get that year? g) When did it reach its low mark? a) 𝑡 7 8 9 10 11 ℎ(𝑡) 31 19 15 19 31 This is in standard form. ℎ = 9, place 9 in the middle of the 𝑥 / 𝑡 𝑐𝑜𝑙𝑢𝑚𝑛. Calculate 𝑦 − 𝑣𝑎𝑙𝑢𝑒𝑠 using a calculator. b) The numbers on the 𝑥 /𝑡 − 𝑎𝑥𝑖𝑠 represent the month of the year, 𝑡 = 1 represents January, 𝑡 = 2 represents February… c) The numbers on 𝑡ℎ𝑒 𝑦 /ℎ(𝑡) − 𝑎𝑥𝑖𝑠 represents the depth of snow at the base of the ski run. d) Vertex (9, 15) e) 𝑥 / 𝑡 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 of the vertex represents the month September. The 𝑦 / ℎ(𝑡) 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 of the vertex represents a 9-foot snow depth at the base of the ski run. f) The lowest snow level was 15 feet. g) The snow depth reached its lowest level in September. Section 6.4: Applications of Quadratic Functions Chapter 6: Quadratic Equations 1) When a ball is thrown straight upward into the air, the equation ℎ(𝑡) = −8𝑡 2 + 80𝑡 gives the height (ℎ(𝑡)) in feet that the ball is above the ground (t) seconds after it is thrown. a) Make a table of values and sketch of graph of the function. b) What do the numbers on the x (horizontal axis) represent? c) What do the numbers on the y (vertical axis) represent? d) Identify the vertex. e) What do the coordinates of the vertex represent? f) When does the ball reach its maximum height? g) What is the maximum height of the ball? h) Find the 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡(𝑠) using Algebra. i) Give a physical interpretation of any 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡. j) Find the practical domain of the function. k) Find the practical range of the function. 2) When an object is thrown straight upward into the air, the equation ℎ(𝑡) = −16𝑡 2 + 64𝑡 gives the height (ℎ(𝑡)) in feet that the ball is above the ground (𝑡) seconds after it is thrown. a) Make a table of values and sketch of graph of the function. b) What do the numbers on the x (horizontal axis) represent? c) What do the numbers on the y (vertical axis) represent? d) Identify the vertex. e) What do the coordinates of the vertex represent? f) When does the ball reach its maximum height? g) What is the maximum height of the ball? h) Find the 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡(𝑠) using Algebra. i) Give a physical interpretation of any 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡. j) Find the practical domain of the function. k) Find the practical range of the function. Section 6.4: Applications of Quadratic Functions Chapter 6: Quadratic Equations 3) An object is launched. The equation for the object's height (ℎ(𝑡)) in meters at time 𝑡 seconds after launch is ℎ(𝑡) = – 16𝑡 2 + 128𝑡 . a) Make a table of values and sketch of graph of the function. b) What do the numbers on the x (horizontal axis) represent? c) What do the numbers on the y (vertical axis) represent? d) Identify the vertex. e) What do the coordinates of the vertex represent? f) When does the ball reach its maximum height? g) What is the maximum height of the ball? h) Find the 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡(𝑠) using Algebra. i) Give a physical interpretation of any 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡. j) Find the practical domain of the function. k) Find the practical range of the function. 4) A golf ball is hit, and its height is given by ℎ(𝑡) = −16𝑡 2 + 96𝑡, where ℎ(𝑡) is its height in meters and t is the time in seconds. a) Make a table of values and sketch of graph of the function. b) What do the numbers on the x (horizontal axis) represent? c) What do the numbers on the y (vertical axis) represent? d) Identify the vertex. e) What do the coordinates of the vertex represent? f) When does the ball reach its maximum height? g) What is the maximum height of the ball? h) Find the 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡(𝑠) using Algebra. i) Give a physical interpretation of any 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡. j) Find the practical domain of the function. k) Find the practical range of the function. Section 6.4: Applications of Quadratic Functions Chapter 6: Quadratic Equations 5) The height h(t) (in feet) of the water level in a reservoir over a 1-year period is modeled by the function. ℎ(𝑡) = 3.3(𝑡 − 9)2 + 14, where 𝑡 = 1 represents January, 𝑡 = 2 represents February, and so on. a) Make a table of values and sketch of graph of the function. b) What do the numbers on the 𝑥 / 𝑡 (ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑎𝑥𝑖𝑠) represent? c) What do the numbers on the 𝑦 / ℎ(𝑡) (𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑎𝑥𝑖𝑠) represent? d) Identify the vertex. e) What do the coordinates of the vertex represent? f) How low did the water level get that year? g) When did it reach its low mark? 6) The height h(t) (in feet) of the water level in a reservoir over a 1-year period is modeled by the function ℎ(𝑡) = 5(𝑡 − 7)2 + 3, where 𝑡 = 1 represents January, 𝑡 = 2 represents February, and so on. a) Make a table of values and sketch of graph of the function. b) What do the numbers on the 𝑥 / 𝑡 (ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑎𝑥𝑖𝑠) represent? c) What do the numbers on the 𝑦 / ℎ(𝑡) (𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑎𝑥𝑖𝑠) represent? d) Identify the vertex. e) What do the coordinates of the vertex represent? f) How low did the water level get that year? g) When did it reach its low mark? 7) The depth (in feet) of the snow at the base of a mountain over a 1-year period is modeled by the function: ℎ(𝑡) = 4(𝑡 − 8)2 + 15, where 𝑡 = 1 represents January, 𝑡 = 2 represents February, and so on. a) Make a table of values and sketch of graph of the function. b) What do the numbers on the 𝑥 / 𝑡 (ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑎𝑥𝑖𝑠) represent? c) What do the numbers on the 𝑦 / ℎ(𝑡) (𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑎𝑥𝑖𝑠) represent? d) Identify the vertex. e) What do the coordinates of the vertex represent? f) How low did the snow level get that year? g) When did it reach its low mark?