Question

when using a quadratic equation in the form $y = ax^2 + bx + c$ to model the height of a projectile ($y$) over time ($x$), which of the following is always represented by the constant term?

• the initial height of the projectile
• the initial velocity of the projectile
• the time at which the projectile hits the ground
• the maximum height of the projectile

Explanation:

To determine what the constant term \( c \) in the quadratic equation \( y = ax^2 + bx + c \) (modeling projectile height \( y \) over time \( x \)) represents, we analyze the meaning of each variable at \( x = 0 \) (initial time, \( t = 0 \)):

• When \( x = 0 \) (initial time), the equation becomes \( y = a(0)^2 + b(0) + c = c \). This \( y \)-value at \( x = 0 \) is the height of the projectile at the start (initial height).
• The initial velocity is related to the coefficient \( b \) (since velocity involves the first - order term in the kinematic model of projectile motion, and in the quadratic equation for height, the coefficient of \( x \) is related to initial velocity).
• The time when the projectile hits the ground is found by solving \( y = 0 \) (i.e., \( ax^2+bx + c = 0 \)) for \( x \), which is not related to the constant term \( c \) directly as a single value for \( c \) doesn't give the time.
• The maximum height is found by using the vertex formula \( x=-\frac{b}{2a} \) and then substituting back into the equation, and it depends on \( a \) and \( b \) as well as \( c \), not just \( c \).

Answer:

the initial height of the projectile