Quadratic Functions Min Max
- 1. Quadratic Functions - Min/Max Maximum Minimum
- 2. What is the maximum value of the quadratic function: 2 y = -2(x + 4) - 9 (-4, -9) What is the minimum value of the quadratic function: 2 y = 3(x - 8) + 14 (8, 14)
- 3. Betty wants to make a rectangle shaped garden. She has 50 meters of fencing. What lengths of sides give Betty a garden with the maximum area? L W 2L + 2W = 50 MAX = LW
- 4. L 2L + 2W = 50 W MAX = LW L = 50 - 2W = 25 - W 2 MAX = (25 - W)W 2 MAX = -W + 25W
- 5. L 2L + 2W = 50 W MAX = LW 2 MAX = -W + 25W 2 MAX = -{W - 25W} 2 2 2 MAX = -{W - 25W + (25/2) - (25/2) } 2 2 MAX = -{(W - 25/2) - (25/2) }
- 6. L 2L + 2W = 50 W MAX = LW 2 2 MAX = -{(W - 25/2) - (25/2) } 2 MAX = -(W - 12.5) + 156.25
- 7. REMINDER: 2 y = a(x - p) + q The value of y is a maximum if a is - The value of y is a maximum when x = - p The value of y is a minimum if a is + The value of y is a minimum when x = -p
- 8. L 2L + 2W = 50 W MAX = LW 2 MAX = -(W - 12.5) + 156.25 - means a maximum does exist The maximum value will be the vertex, or when W = 12.5, it is 156.25.
- 9. L 2L + 2W = 50 W MAX = LW The maximum value will be the vertex, or when W = 12.5, it is 156.25. Can substitute these values back into the original equations to find the value of L 2L + 2(12.5) = 50 L = 50 - 25 = 12.5 2
- 10. We calculated that the maximum area that can be enclosed with 50 meters 2 of fence is 156.25 m . This area is possible when each side is 12.5m long. 12.5m 12.5m 2 12.5m 156.25m 12.5m
- 11. 304 people will go to a basketball game if the tickets cost 8 dollars. Every time the price is increase $0.50 16 fewer people will go to the game. What ticket price gives the maximum profits?
- 12. Profit = (Number of people)*(Cost per person) P=N*C x = number of times the price is increased P =N *C m m m Nm = N - 16x Cm = C + .5x Pm = (N - 16x) (C + .5x)
- 13. Pm = (N - 16x) (C + .5x) Pm = (304 - 16x) (8 + .5x) P = 2432 + 152x - 128x -8x2 m Pm = -8x2 + 24x + 2432
- 14. 2 + 24x + 2432 Pm = -8x 2 - 3x} Pm = -8{x + 2432 2 - 3x + 1.52 - 1.52} +2432 Pm = -8{x 2 2 Pm = -8{(x - 1.5) -1.5 } + 2432 Pm = -8(x-1.5) 2 + 18 + 2432
- 15. P = -8(x-1.5) 2 + 18 + 2432 m P = -8(x-1.5) 2 + 2450 m So, the max profit is when x = 1.5 x = number of times the price is increased Cm = C + .5x Cm = 8 + .5 (1.5) Cm = $8.75
- 16. ex. 6 1-5