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Linear Systems, Quadratics - Coggle Diagram

- - - - The super-car Auto shop charges an $50 initial fee and a $20 daily fee. The Fix-up Auto shop charges an $100 initial fee and a $10 daily fee.
        20d+50=10d+100
        20d-10d=100-50
        10d/10d=50/10=5
        C=20(5)+50
        C=100+50
        C=150
        C=10(5)+100
        C=50+100
        C=150
        
        After 5 days the Fixup AutoShop is cheaper to pay for.
- - - - Add and subtract the C term
      - Move the negative value out the bracket and multiply by the number in front of the bracket.
      - The x² should be 1. Factor the greatest number form the a and b values.
      - Factor new trinomial in the brackets.
      - To create the C term take the b value and half it then square it
      - New equation is in vertex form
      - y = x² + 16x + 120
        c=(b/2)²
        c=(16/2)²
        c=(8)²
        c=64
        y = (x² + 16x + 64 - 64) + 120
        y = (x² + 16x + 64) + 120 - 64
        y = (x² + 16x + 64) + 56
        y = (x + 8)² + 56
        Vertex: (-8, 56)
        AoS: -8
        Max: 56
  - - - Standard for can be used to find factored form by putting the values of a, b and c into the quadratic formula
        
        y = 3x² - 5x - 8
        x = -(-5) +- (-5)² -4(3)(-8)
        2(3)
        x = 5+- 25 + 96
        6
        x = 5+- 121
        6
        x = 5 + 121
        
        6
        x = 2.7
        x= 5 - 121
        6
        x = 1
- - - - Devin Booker takes a 3 point shot during the final match with the Lakers. The basketballs path is shown by the equation h = -t² + 6x + 15. h is the height of the ball and x is time in seconds after the ball was shot. What is the maximum height the ball reaches?
        h = -t² + 6x + 15
        h = -(0)² + 6(0) + 15
        h = 0 + 0 + 15
        h = 15
        h = -(1)² - 6(1) + 15
        h = -1 + 6 + 15
        h = 20
        The maximum height the ball reaches is 24m.
    - - A table with differences is used to find out if an equation is a quadratic one. If there is a consistent second difference than it is a quadratic.
  - - - y = 3x² + 5
        y = 3(-2)² + 5
        y = 12 + 5
        y = 17
        y = 3(-1)² + 5
        y = 3 + 5
        y = 8
        y = 3(0)² + 5
        y = 0 + 5
        y = 5
        y = 3(1)² + 5
        y = 3 + 5
        y = 8
        y = 3(2)² + 5
        y = 12 + 5
        y = 17
        y = 3(3)² + 5
        y = 27 + 5
        y = 32
    - - y = 3(x-2)²
        y = 3(-2-2)²
        y = 3(-4)²
        y = 3(16)
        y = 48
        y = 3(-1-2)²
        y = 3(-3)²
        y = 3(9)
        y = 27
        y = 3(0-2)²
        y = 3(-2)²
        y = 3(4)
        y = 12
        y = 3(1-2)²
        y = 3(-1)²
        y = 3(1)
        y = 3
        y = 3(2-2)²
        y = 3(0)²
        y = 3(0)
        y = 0
  - - - Chris Paul shoots a basketball at the starting point of 5m off the ground and reaches a maximum point of 25m in height of the ground at 15m away from Chris Paul. Let h represent the height and x represent the horizontal distance. Make an equation to figure out the path of the ball. If the net is 20m away at what height will Chris Paul's teammate get the rebound.
        h = a(x - h)² + k
        5 = a(0 - 25)² + 15
        5 = a(-25)² +15
        5 - 15 = 625a +15-15
        -10/625 = 625a/625a
        -0.016 = a
        h = -0.016(20 - 25)² + 15
        h = -0.016(-5)² + 15
        h = -0.016(25) + 15
        h = 14.6
  - - - Jordan is checking out his physical skills. He wants to see how high he can jump and how far he can jump. What is the distance of Jordan’s jump? What is the maximum height of Jordan’s jump? d represents the distance of the ground. h represents the height of the jump. Jordan can jump 20 cm and can jump 100cm high.
        AoS = 5 + 25
        AoS = 30/2
        AoS = 15
        h = -1(d - 5)(d - 25)
        h = -1(15 - 5)( 15 - 25)
        h = -1(10)(-10)
        h = -1(-100)
        h = 100
- - - - Trinomial as a result of squaring a binomial
    - - Binomial that comes from multiplying 2 binomials with equal values but different operation
  - - - (x - 5)(x + 6)
        x² + 6x - 5x - 30
        x² + 1x - 30