Flashcardizz: 3-4 Modeling with Quadratic Functions (Gravity)

A quadratic function is a polynomial function of degree 2, typically in the form of $$f(x) = ax^2 + bx + c$$ , where a, b, and c are constants and a ≠ 0.

The vertex of a quadratic function represents the maximum or minimum point of the graph, depending on the direction of the parabola.

The maximum height can be found at the vertex of the parabola, which occurs at $$t = -\frac{b}{2a}$$ for the function $$h(t) = at^2 + bt + c$$ .

The coefficient 'a' determines the direction of the parabola: if 'a' is positive, the parabola opens upwards; if 'a' is negative, it opens downwards.

The initial height is the value of h when t = 0, represented by the constant term c in the quadratic equation $$h(t) = at^2 + bt + c$$ .

The time to reach maximum height can be calculated using the formula $$t = -\frac{b}{2a}$$ from the quadratic equation $$h(t) = at^2 + bt + c$$ .

The height of an object in free fall can be modeled by the equation $$h(t) = -16t^2 + v_ot + h_o$$ , where v_o is the initial velocity and h_o is the initial height.

In the context of quadratic functions modeling motion, gravity is represented by the coefficient -16 in the equation, which accounts for the acceleration due to gravity in feet per second squared.

The y-intercept is the point where the graph intersects the y-axis, representing the initial height of the object when time t = 0.

To calculate the height at a specific time, substitute the value of t into the quadratic equation $$h(t) = at^2 + bt + c$$ .