Certainly! Let's look at the quadratic equation used to model the height [tex]\( y \)[/tex] of a projectile over time [tex]\( x \)[/tex]:
[tex]\[ y = ax^2 + bx + c \][/tex]
In this equation:
- [tex]\( a \)[/tex] represents the coefficient related to the acceleration due to gravity divided by 2 (usually a negative number because gravity is pulling the projectile downwards).
- [tex]\( b \)[/tex] represents the initial velocity of the projectile.
- [tex]\( c \)[/tex] is the constant term.
To understand what the constant term [tex]\( c \)[/tex] represents, we can examine the equation at time [tex]\( x = 0 \)[/tex].
When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = a(0)^2 + b(0) + c \][/tex]
[tex]\[ y = 0 + 0 + c \][/tex]
[tex]\[ y = c \][/tex]
This shows that when [tex]\( x = 0 \)[/tex], the height [tex]\( y \)[/tex] of the projectile is equal to [tex]\( c \)[/tex]. This means that [tex]\( c \)[/tex] is the initial height of the projectile when it has just been launched (i.e., before it has started moving due to its initial velocity or the influence of gravity).
So, the constant term [tex]\( c \)[/tex] in the quadratic equation [tex]\( y = ax^2 + bx + c \)[/tex] represents the initial height of the projectile.
Therefore, the correct interpretation of the constant term is:
- the initial height of the projectile.
