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

A. The initial height of the projectile
B. The initial velocity of the projectile
C. The time at which the projectile hits the ground
D. The maximum height of the projectile

Sagot :

GinnyAnswer
Sure, let's work through the details step-by-step.

A quadratic equation is often used to model the height of a projectile. Such an equation is typically written in the form:

[tex]\[ y = ax^2 + bx + c \][/tex]

Here, [tex]\(y\)[/tex] represents the height of the projectile at any time [tex]\(x\)[/tex]. Now let's analyze each term in this expression:

1. Term [tex]\(ax^2\)[/tex]:
- This term involves [tex]\(x\)[/tex] squared and typically represents the influence of acceleration on the projectile. In the context of projectile motion, this is usually due to gravity.

2. Term [tex]\(bx\)[/tex]:
- This term represents the linear component of the motion. It combines with time [tex]\(x\)[/tex] and often represents the initial velocity of the projectile in the context of projectile motion.

3. Constant term [tex]\(c\)[/tex]:
- This term does not involve [tex]\(x\)[/tex] and thus is independent of time.

To understand the constant term [tex]\(c\)[/tex] more specifically:

- When we input [tex]\(x = 0\)[/tex] into the quadratic equation (which corresponds to the initial time):
[tex]\[ y = a(0)^2 + b(0) + c \][/tex]
[tex]\[ y = c \][/tex]
- Therefore, [tex]\(c\)[/tex] represents the value of [tex]\(y\)[/tex] when [tex]\(x = 0\)[/tex]. In the context of projectile motion, this means that [tex]\(c\)[/tex] is the initial height of the projectile.

Now, let's analyze each of the given options:

1. The initial height of the projectile:
- This option is correct, as we have determined that the constant term [tex]\(c\)[/tex] in the quadratic equation represents the height of the projectile when [tex]\(x = 0\)[/tex], which is the initial height.

2. The initial velocity of the projectile:
- This is represented by the coefficient [tex]\(b\)[/tex] of the term [tex]\(bx\)[/tex].

3. The time at which the projectile hits the ground:
- This is not directly represented by any single term in the quadratic equation but can be found by solving the equation for when [tex]\(y = 0\)[/tex].

4. The maximum height of the projectile:
- This is found by analyzing the vertex of the parabola represented by the quadratic equation, not by the constant term.

Hence, the correct and always-represented-by-the-constant-term answer is:

The initial height of the projectile.
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