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Quadratics - Coggle Diagram

- - - - Shapes a parabola
        
        From this we can get the Vertex, AoS and the opening of the Parabola
        
        Vertex: The maximum or minimum value of a parabola
        Axis of Symmetry (AoS): the x value of the middle of the parabola
        Opening of parabola: This is the way the parabola is curved. If it's a "n" shape then its opening downward. If it's a "u" shape then it's opening upward
- - - - The path of a tennis ball that was hit can be represented by the equation h = -s^2 + 5s + 10 where h is the height of the ball in meters above the ground and s is the time in seconds after the ball was hit. What was the height of the ball after 1 second?
        s = 1
        h = -(1)^2 + 5(1) + 10
        h = -1 + 5 + 10
        h = 14
        Therefore the height of the ball after 1 second is 14m
- - - - A shoe company models the profit on the latest shoes using the relation, P = -5x^2 + 46x - 9, where x is the number of shoes produced in hundred thousands and P is profit in millions of dollars. Factor the following equation and determind where the break-even points are.
        P = -5x^2 + 46x - 9
        P = -5x^2 + 1x + 45x - 9
        P = (-5x^2 + 1x)(+ 45x - 9)
        P = (-5x^2 + 1x)-1x(+ 45x - 9)/9
        P = -1x(5x - 1) +9(5x - 1)
        P = (5x - 1)(-1x + 9)
        5x - 1 = 0
        5x = 1
        5x/5 = 1/5
        x = 0.2
        -1x + 9 = 0
        9 = 1x
        9/1 = 1x/1
        9 = x
        Therefore break even at 20,000 pairs of shoes & 900,000 pairs of shoes
- - - - Question: A toaster company has a profit, P, that can be modelled by P = -5000x^2 + 75000x + 200000, where x is the cost of the toasters that they manufacture. What should the company charge to maximize profit?
        Complete the Square
        P = (-5000x^2/-5000 + 75000x/-5000) + 200000
        P = -5000(x^2 - 17) + 200000
        C = (15/2)^2
        C = (7.5)^2
        C = 56.25
        P = -5000(x^2 - 15 + 56.25 - 56.25) + 200000
        P = -5000(x^2 - 15 + 56.25) + 200000 - (-56.25 x -5000)
        P = -5000(x^2 - 15 + 56.25) + 481250
        P = -5000(x - 56.25)^2 + 481250
        Therefore the company should charge $56.25 for maximum profit