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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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**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**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.**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)**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 ) :**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 ) :**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**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 )**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.**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.**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**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**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**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**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**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**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.**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). .**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 )**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**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**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**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**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**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**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”**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”**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”**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 )**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 )**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…**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**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”…**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**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**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**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**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”**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**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**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**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.**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.**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**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 :**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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