Question 282339: To describe a trajectory of a missile or a football, we use quadratic functions like y(t)=a*t^2+b*t+c, where y is the height after time t sec. From your experience of seeing the path of football or other projectiles, what are the limitation on the parameters a, b and c? If the function is describing the football path t seconds after it left the player's hand, what does c mean?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 
In the graphs above:
The coefficient of the x term is 0.
The coefficient of the constant term is 6.
The coefficients of the x^2 terms are -.001, -1, -100.
The more negative the coefficient of the x^2 term is, the steeper the trajectory of the parabola and the narrower its width.
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In the graphs above:
The coefficient of the x^2 term is 1.
The coefficient of the constant term is 6.
The coefficients of the x terms are -5, -10, -15.
The more negative the coefficient of the x term gets, the more to the left the maximum value of the equation gets, and the higher the maximum value of the equation gets.
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In the graphs above:
The coefficient of the x^2 term is 1.
The coefficient of the constant term is 6.
The coefficients of the x terms are 5, 10, 15.
The more positive the coefficient of the x term gets, the more to the right the maximum value of the equation gets, and the higher the maximum value of the equation gets.
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Impact on using the quadratic equation to describing the trajectory of the football after x second (x is used instead of t in order to show the relationship on the graph).
The standard form of the quadratic equation is ax^2 + bx + c.
a is the coefficient of the x^2 term and has to be negative.
This allows the trajectory of the football to follow an arc to reach a maximum point and then fall back to earth.
b is the coefficient of the x^2 term and can be negative or positive.
If b is more negative, the high point of the arc of the football shifts to the left.
If b is positive, the high point of the arc of the football shifts to the right.
Even though the high point of the arc of the football travel can be to the left of the y-axis, some of the arc of the football will be to the right of the y-axis which would be the period of time of interest and is therefore valid.
This happens when c is greater than 0 as it has to be.
c is the constant term and has to be positive and should be approximately the height of the football player who is throwing the football.
c represents the height of the football when t = 0 which is the point at which the ball is thrown.
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You start off with c = the height of the football player who is throwing the football.
a has to be negative.
The more negative it is, the steeper the trajectory of the football, but this has no impact on the height of the trajectory. As long as b equals 0, the height will be the value of c.
b can be negative or positive.
The value of b affects the height of the trajectory and whether or not the maximum point of the arc of the football will be shifted to the left or the right.
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