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FUNCTIONS : Solving Problems with Linear / Quadratic Functions LINEAR RELATIONSHIP - a direct proportion with x and

FUNCTIONS : Solving Problems with Linear / Quadratic Functions LINEAR RELATIONSHIP - a direct proportion with x and y - use slope – intercept form ( y = mx + b ) - find the slope ( m ) first with the given information

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FUNCTIONS : Solving Problems with Linear / Quadratic Functions LINEAR RELATIONSHIP - a direct proportion with x and

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  1. FUNCTIONS : Solving Problems with Linear / Quadratic Functions LINEAR RELATIONSHIP - a direct proportion with x and y - use slope – intercept form ( y = mx + b ) - find the slope ( m ) first with the given information - find the y – intercept ( b ) using one of the given points

  2. FUNCTIONS : Solving Problems with Linear / Quadratic Functions LINEAR RELATIONSHIP - a direct proportion with x and y - use slope – intercept form ( y = mx + b ) - find the slope ( m ) first with the given information - find the y – intercept ( b ) using one of the given points EXAMPLE : Whitewater Rafters’ revenue function is a linear function of the number of people, x , who go rafting on a weekend. If 300 people go, the revenue is $900.00. If 500 people go, the revenue is $1400.00. a) Find the function r (x), which represents the revenue from the number of people who go rafting during one weekend. b) Find the revenue when 450 people go rafting.

  3. FUNCTIONS : Solving Problems with Linear / Quadratic Functions LINEAR RELATIONSHIP - a direct proportion with x and y - use slope – intercept form ( y = mx + b ) - find the slope ( m ) first with the given information - find the y – intercept ( b ) using one of the given points EXAMPLE : Whitewater Rafters’ revenue function is a linear function of the number of people, x , who go rafting on a weekend. If 300 people go, the revenue is $900.00. If 500 people go, the revenue is $1400.00. a) Find the function r (x), which represents the revenue from the number of people who go rafting during one weekend. b) Find the revenue when 450 people go rafting. Given information : ( 300, 900 ) ( 500, 1400)

  4. FUNCTIONS : Solving Problems with Linear / Quadratic Functions LINEAR RELATIONSHIP - a direct proportion with x and y - use slope – intercept form ( y = mx + b ) - find the slope ( m ) first with the given information - find the y – intercept ( b ) using one of the given points EXAMPLE : Whitewater Rafters’ revenue function is a linear function of the number of people, x , who go rafting on a weekend. If 300 people go, the revenue is $900.00. If 500 people go, the revenue is $1400.00. a) Find the function r (x), which represents the revenue from the number of people who go rafting during one weekend. b) Find the revenue when 450 people go rafting. Given information : ( 300, 900 ) ( 500, 1400 ) First, find the slope ( m ) :

  5. FUNCTIONS : Solving Problems with Linear / Quadratic Functions LINEAR RELATIONSHIP - a direct proportion with x and y - use slope – intercept form ( y = mx + b ) - find the slope ( m ) first with the given information - find the y – intercept ( b ) using one of the given points EXAMPLE : Whitewater Rafters’ revenue function is a linear function of the number of people, x , who go rafting on a weekend. If 300 people go, the revenue is $900.00. If 500 people go, the revenue is $1400.00. a) Find the function r (x), which represents the revenue from the number of people who go rafting during one weekend. b) Find the revenue when 450 people go rafting. Given information : ( 300, 900 ) ( 500, 1400 ) First, find the slope ( m ) :

  6. FUNCTIONS : Solving Problems with Linear / Quadratic Functions LINEAR RELATIONSHIP - a direct proportion with x and y - use slope – intercept form ( y = mx + b ) - find the slope ( m ) first with the given information - find the y – intercept ( b ) using one of the given points EXAMPLE : Whitewater Rafters’ revenue function is a linear function of the number of people, x , who go rafting on a weekend. If 300 people go, the revenue is $900.00. If 500 people go, the revenue is $1400.00. a) Find the function r (x), which represents the revenue from the number of people who go rafting during one weekend. b) Find the revenue when 450 people go rafting. Substitute m into y = mx +b

  7. FUNCTIONS : Solving Problems with Linear / Quadratic Functions LINEAR RELATIONSHIP - a direct proportion with x and y - use slope – intercept form ( y = mx + b ) - find the slope ( m ) first with the given information - find the y – intercept ( b ) using one of the given points EXAMPLE : Whitewater Rafters’ revenue function is a linear function of the number of people, x , who go rafting on a weekend. If 300 people go, the revenue is $900.00. If 500 people go, the revenue is $1400.00. a) Find the function r (x), which represents the revenue from the number of people who go rafting during one weekend. b) Find the revenue when 450 people go rafting. Substituted the given point ( 300, 900 )

  8. FUNCTIONS : Solving Problems with Linear / Quadratic Functions LINEAR RELATIONSHIP - a direct proportion with x and y - use slope – intercept form ( y = mx + b ) - find the slope ( m ) first with the given information - find the y – intercept ( b ) using one of the given points EXAMPLE : Whitewater Rafters’ revenue function is a linear function of the number of people, x , who go rafting on a weekend. If 300 people go, the revenue is $900.00. If 500 people go, the revenue is $1400.00. a) Find the function r (x), which represents the revenue from the number of people who go rafting during one weekend. b) Find the revenue when 450 people go rafting.

  9. FUNCTIONS : Solving Problems with Linear / Quadratic Functions LINEAR RELATIONSHIP - a direct proportion with x and y - use slope – intercept form ( y = mx + b ) - find the slope ( m ) first with the given information - find the y – intercept ( b ) using one of the given points EXAMPLE : Whitewater Rafters’ revenue function is a linear function of the number of people, x , who go rafting on a weekend. If 300 people go, the revenue is $900.00. If 500 people go, the revenue is $1400.00. a) Find the function r (x), which represents the revenue from the number of people who go rafting during one weekend. b) Find the revenue when 450 people go rafting.

  10. FUNCTIONS : Solving Problems with Linear / Quadratic Functions LINEAR RELATIONSHIP - a direct proportion with x and y - use slope – intercept form ( y = mx + b ) - find the slope ( m ) first with the given information - find the y – intercept ( b ) using one of the given points EXAMPLE : Whitewater Rafters’ revenue function is a linear function of the number of people, x , who go rafting on a weekend. If 300 people go, the revenue is $900.00. If 500 people go, the revenue is $1400.00. a) Find the function r (x), which represents the revenue from the number of people who go rafting during one weekend. b) Find the revenue when 450 people go rafting. Substitute m & b into y = mx + b

  11. FUNCTIONS : Solving Problems with Linear / Quadratic Functions LINEAR RELATIONSHIP - a direct proportion with x and y - use slope – intercept form ( y = mx + b ) - find the slope ( m ) first with the given information - find the y – intercept ( b ) using one of the given points EXAMPLE : Whitewater Rafters’ revenue function is a linear function of the number of people, x , who go rafting on a weekend. If 300 people go, the revenue is $900.00. If 500 people go, the revenue is $1400.00. a) Find the function r (x), which represents the revenue from the number of people who go rafting during one weekend. b) Find the revenue when 450 people go rafting. This is your revenue function

  12. FUNCTIONS : Solving Problems with Linear / Quadratic Functions LINEAR RELATIONSHIP - a direct proportion with x and y - use slope – intercept form ( y = mx + b ) - find the slope ( m ) first with the given information - find the y – intercept ( b ) using one of the given points EXAMPLE : Whitewater Rafters’ revenue function is a linear function of the number of people, x , who go rafting on a weekend. If 300 people go, the revenue is $900.00. If 500 people go, the revenue is $1400.00. a) Find the function r (x), which represents the revenue from the number of people who go rafting during one weekend. b) Find the revenue when 450 people go rafting. Substitute 450 for x into the revenue function

  13. FUNCTIONS : Solving Problems with Linear / Quadratic Functions LINEAR RELATIONSHIP - a direct proportion with x and y - use slope – intercept form ( y = mx + b ) - find the slope ( m ) first with the given information - find the y – intercept ( b ) using one of the given points EXAMPLE : Whitewater Rafters’ revenue function is a linear function of the number of people, x , who go rafting on a weekend. If 300 people go, the revenue is $900.00. If 500 people go, the revenue is $1400.00. a) Find the function r (x), which represents the revenue from the number of people who go rafting during one weekend. b) Find the revenue when 450 people go rafting. Substitute 450 for x into the revenue function

  14. FUNCTIONS : Solving Problems with Linear / Quadratic Functions LINEAR RELATIONSHIP - a direct proportion with x and y - use slope – intercept form ( y = mx + b ) - find the slope ( m ) first with the given information - find the y – intercept ( b ) using one of the given points EXAMPLE : Whitewater Rafters’ revenue function is a linear function of the number of people, x , who go rafting on a weekend. If 300 people go, the revenue is $900.00. If 500 people go, the revenue is $1400.00. a) Find the function r (x), which represents the revenue from the number of people who go rafting during one weekend. b) Find the revenue when 450 people go rafting. $1,275.00

  15. FUNCTIONS : Solving Problems with Linear / Quadratic Functions QUADRATIC RELATIONSHIP - not a proportional relationship - as x increases proportionally, y increases faster - use quadratic form y = ax2 + bx + c

  16. FUNCTIONS : Solving Problems with Linear / Quadratic Functions QUADRATIC RELATIONSHIP - not a proportional relationship - as x increases proportionally, y increases faster - use quadratic form y = ax2 + bx + c Let’s dive right into an example …note the steps.

  17. FUNCTIONS : Solving Problems with Linear / Quadratic Functions EXAMPLE : Camera Cases Ltd. Produces camera cases. They have found that the cost, c(x), of making x camera cases is a quadratic function in terms of x. The company knows that it costs $23 to produce 2 camera cases, $103 to produce 4 camera cases, and $631 to produce 10 camera cases. Find Camera Cases Ltd. Cost function c(x). .

  18. FUNCTIONS : Solving Problems with Linear / Quadratic Functions EXAMPLE : Camera Cases Ltd. Produces camera cases. They have found that the cost, c(x), of making x camera cases is a quadratic function in terms of x. The company knows that it costs $23 to produce 2 camera cases, $103 toproduce 4 camera cases, and $631 to produce 10 camera cases. Find Camera Cases Ltd. Cost function c(x). STEP 1 - Given info : ( 2 , 23 ) ( 4 , 103 ) ( 10 , 631 )

  19. FUNCTIONS : Solving Problems with Linear / Quadratic Functions EXAMPLE : Camera Cases Ltd. Produces camera cases. They have found that the cost, c(x), of making x camera cases is a quadratic function in terms of x. The company knows that it costs $23 to produce 2 camera cases, $103 to produce 4 camera cases, and $631 to produce 10 camera cases. Find Camera Cases Ltd. Cost function c(x). STEP 1 - Given info : ( 2 , 23 ) ( 4 , 103 ) ( 10 , 631 ) STEP 2 – plug the given info into the quadratic function

  20. FUNCTIONS : Solving Problems with Linear / Quadratic Functions EXAMPLE : Camera Cases Ltd. Produces camera cases. They have found that the cost, c(x), of making x camera cases is a quadratic function in terms of x. The company knows that it costs $23 to produce 2 camera cases, $103 to produce 4 camera cases, and $631 to produce 10 camera cases. Find Camera Cases Ltd. Cost function c(x). STEP 1 - Given info : ( 2 , 23 ) ( 4 , 103 ) ( 10 , 631 ) STEP 2 – plug the given info into the quadratic function

  21. FUNCTIONS : Solving Problems with Linear / Quadratic Functions EXAMPLE : Camera Cases Ltd. Produces camera cases. They have found that the cost, c(x), of making x camera cases is a quadratic function in terms of x. The company knows that it costs $23 to produce 2 camera cases, $103 to produce 4 camera cases, and $631 to produce 10 camera cases. Find Camera Cases Ltd. Cost function c(x). STEP 1 - Given info : ( 2 , 23 ) ( 4 , 103 ) ( 10 , 631 ) STEP 2 – plug the given info into the quadratic function

  22. FUNCTIONS : Solving Problems with Linear / Quadratic Functions So far we have : Equation #1 : Equation #2 : Equation #3 : STEP 3 : Eq. #2 – Eq. #1 103 = 16a + 4b + c – 23 = 4a + 2b + c 80 = 12a + 2b

  23. FUNCTIONS : Solving Problems with Linear / Quadratic Functions So far we have : Equation #1 : Equation #2 : Equation #3 : STEP 3 : Eq. #2 – Eq. #1 103 = 16a + 4b + c – 23 = 4a + 2b + c 80 = 12a + 2b - this becomes Eq. # 4

  24. FUNCTIONS : Solving Problems with Linear / Quadratic Functions So far we have : Equation #1 : Equation #2 : Equation #3 : STEP 3 : Eq. #2 – Eq. #1 103 = 16a + 4b + c – 23 = 4a + 2b + c 80 = 12a + 2b Eq. # 3 – Eq. # 2 631 = 100a +10b + c – 103 = 16a + 4b + c 528 = 84a + 6b - this becomes Eq. # 4 - this becomes Eq. # 5

  25. FUNCTIONS : Solving Problems with Linear / Quadratic Functions So far we have : Equation #1 : Equation #2 : Equation #3 : STEP 3 : Eq. #2 – Eq. #1 103 = 16a + 4b + c – 23 = 4a + 2b + c 80 = 12a + 2b Eq. # 3 – Eq. # 2 631 = 100a +10b + c – 103 = 16a + 4b + c 528 = 84a + 6b - this becomes Eq. # 4 - this becomes Eq. # 5 ** As you can see, step 3 gets rid of our variable “c”

  26. FUNCTIONS : Solving Problems with Linear / Quadratic Functions So far we have : Equation #1 : Equation #2 : Equation #3 : STEP 3 : Eq. #2 – Eq. #1 103 = 16a + 4b + c – 23 = 4a + 2b + c 80 = 12a + 2b Eq. # 3 – Eq. # 2 631 = 100a +10b + c – 103 = 16a + 4b + c 528 = 84a + 6b - this becomes Eq. # 4 - this becomes Eq. # 5 ** As you can see, step 3 gets rid of our variable “c” - now we will use Eq’s 4 & 5 to get rid of “b” and solve for “a”

  27. FUNCTIONS : Solving Problems with Linear / Quadratic Functions So far we have : Equation #1 : Equation #2 : Equation #3 : STEP 4 : 528 = 84a + 6b + 80 = 12a + 2b Eq. # 5 Eq. # 4 ** we can now use the addition method for systems of equations to eliminate “b” and solve for “a”

  28. FUNCTIONS : Solving Problems with Linear / Quadratic Functions So far we have : Equation #1 : Equation #2 : Equation #3 : STEP 4 : 528 = 84a + 6b + (-3) (80 = 12a + 2b) Eq. # 5 Eq. # 4 1st – multiply Eq. #4 by ( - 3 )

  29. FUNCTIONS : Solving Problems with Linear / Quadratic Functions So far we have : Equation #1 : Equation #2 : Equation #3 : STEP 4 : 528 = 84a + 6b + -240 = -36a - 6b Eq. # 5 Eq. # 4 1st – multiply Eq. #4 by ( - 3 )

  30. FUNCTIONS : Solving Problems with Linear / Quadratic Functions So far we have : Equation #1 : Equation #2 : Equation #3 : STEP 4 : 528 = 84a + 6b + -240 = -36a - 6b Eq. # 5 Eq. # 4 288 = 48a Use addition method…

  31. FUNCTIONS : Solving Problems with Linear / Quadratic Functions So far we have : Equation #1 : Equation #2 : Equation #3 : STEP 4 : 528 = 84a + 6b + -300 = -36a - 6b Eq. # 5 Eq. # 4 288 = 48a a = 6

  32. FUNCTIONS : Solving Problems with Linear / Quadratic Functions So far we have : Equation #1 : Equation #2 : Equation #3 : a = 6 STEP 5 : Plug a = 6 into Eq. # 4 and solve for “b”…

  33. FUNCTIONS : Solving Problems with Linear / Quadratic Functions So far we have : Equation #1 : Equation #2 : Equation #3 : a = 6 STEP 5 : Plug a = 6 into Eq. # 4 and solve for “b”… 80 = 12a + 2b Eq. # 4 80 = 12(6) + 2b

  34. FUNCTIONS : Solving Problems with Linear / Quadratic Functions So far we have : Equation #1 : Equation #2 : Equation #3 : a = 6 STEP 5 : Plug a = 6 into Eq. # 4 and solve for “b”… 80 = 12a + 2b Eq. # 4 80 = 12(6) + 2b 80 = 72 + 2b -72 = -72

  35. FUNCTIONS : Solving Problems with Linear / Quadratic Functions So far we have : Equation #1 : Equation #2 : Equation #3 : a = 6 STEP 5 : Plug a = 6 into Eq. # 4 and solve for “b”… 80 = 12a + 2b Eq. # 4 80 = 12(6) + 2b 80 = 72 + 2b -72 = -72 8 = 2b b = 4

  36. FUNCTIONS : Solving Problems with Linear / Quadratic Functions So far we have : Equation #1 : Equation #2 : Equation #3 : a = 6 b = 4 STEP 5 : Plug a = 6 into Eq. # 4 and solve for “b”… 80 = 12a + 2b Eq. # 4 80 = 12(6) + 2b 80 = 72 + 2b -72 = -72 8 = 2b b = 4

  37. FUNCTIONS : Solving Problems with Linear / Quadratic Functions So far we have : Equation #1 : Equation #2 : Equation #3 : a = 6 b = 4 STEP 5 : Plug a = 6 , b = 4 into Eq. # 1 and solve for “c”

  38. FUNCTIONS : Solving Problems with Linear / Quadratic Functions So far we have : Equation #1 : Equation #2 : Equation #3 : a = 6 b = 4 STEP 5 : Plug a = 6 , b = 4 into Eq. # 1 and solve for “c” 23 = 4a + 2b + c 23 = 4(6) + 2(4) + c

  39. FUNCTIONS : Solving Problems with Linear / Quadratic Functions So far we have : Equation #1 : Equation #2 : Equation #3 : a = 6 b = 4 STEP 5 : Plug a = 6 , b = 4 into Eq. # 1 and solve for “c” 23 = 4a + 2b + c 23 = 4(6) + 2(4) + c 23 = 24 + 8 + c 23 = 32 + c c = -9

  40. FUNCTIONS : Solving Problems with Linear / Quadratic Functions So far we have : Equation #1 : Equation #2 : Equation #3 : a = 6 b = 4 c = - 9 STEP 5 : Plug a = 6 , b = 4 into Eq. # 1 and solve for “c” 23 = 4a + 2b + c 23 = 4(6) + 2(4) + c 23 = 24 + 8 + c 23 = 32 + c c = -9

  41. FUNCTIONS : Solving Problems with Linear / Quadratic Functions So far we have : Equation #1 : Equation #2 : Equation #3 : a = 6 b = 4 c = - 9 STEP 6 : Now we plug a = 6, b = 4 , and c = - 9 into c(x) = ax2 + bx + c and it becomes our cost function for Camera Cases Ltd.

  42. FUNCTIONS : Solving Problems with Linear / Quadratic Functions EXAMPLE : Camera Cases Ltd. Produces camera cases. They have found that the cost, c(x), of making x camera cases is a quadratic function in terms of x. The company knows that it costs $23 to produce 2 camera cases, $103 to produce 4 camera cases, and $631 to produce 10 camera cases. Find Camera Cases Ltd. Cost function c(x). a = 6 b = 4 c = - 9 STEP 6 : Now we plug a = 6, b = 4 , and c = - 9 into c(x) = ax2 + bx + c and it becomes our cost function for Camera Cases Ltd.

  43. FUNCTIONS : Solving Problems with Linear / Quadratic Functions EXAMPLE : Camera Cases Ltd. Produces camera cases. They have found that the cost, c(x), of making x camera cases is a quadratic function in terms of x. The company knows that it costs $23 to produce 2 camera cases, $103 to produce 4 camera cases, and $631 to produce 10 camera cases. Find Camera Cases Ltd. Cost function c(x). a = 6 b = 4 c = - 9 STEP 6 : Now we plug a = 6, b = 4 , and c = - 9 into c(x) = ax2 + bx + c and it becomes our cost function for Camera Cases Ltd. c(x) = 6x2 + 4x – 9

  44. FUNCTIONS : Solving Problems with Linear / Quadratic Functions EXAMPLE : Camera Cases Ltd. Produces camera cases. They have found that the cost, c(x), of making x camera cases is a quadratic function in terms of x. The company knows that it costs $23 to produce 2 camera cases, $103 to produce 4 camera cases, and $631 to produce 10 camera cases. Find Camera Cases Ltd. Cost function c(x). STEP 6 : Now we plug a = 6, b = 4 , and c = - 9 into c(x) = ax2 + bx + c and it becomes our cost function for Camera Cases Ltd. c(x) = 6x2 + 4x – 9 To find the cost of making 25 cameras :

  45. FUNCTIONS : Solving Problems with Linear / Quadratic Functions EXAMPLE : Camera Cases Ltd. Produces camera cases. They have found that the cost, c(x), of making x camera cases is a quadratic function in terms of x. The company knows that it costs $23 to produce 2 camera cases, $103 to produce 4 camera cases, and $631 to produce 10 camera cases. Find Camera Cases Ltd. Cost function c(x). STEP 6 : Now we plug a = 6, b = 4 , and c = - 9 into c(x) = ax2 + bx + c and it becomes our cost function for Camera Cases Ltd. c(x) = 6x2 + 4x – 9 To find the cost of making 25 cameras : c(x) = 6(25)2 + 4(25) – 9 c(x) = 6(625) + 100 – 9 c(x) = 3750 + 100 – 9 c(x) = 3841

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