Question

the height y, in feet, of a football x seconds after it is thrown is modeled by the equation \\(y = -16x^2 + 24x + 1\\). what information does the zero of the equation \\(6 = -16x^2 + 24x + 1\\) represent?

the height of the ball after 6 seconds
when the ball is 6 ft above the ground
the initial upward velocity of the ball
the horizontal distance traveled by the ball when it is 6 ft above the ground

Explanation:

Identify the variables in the model

The given equation is \(y = -16x^2 + 24x + 1\).
The variable \(y\) represents the height of the football in feet.
The variable \(x\) represents the time in seconds after the football is thrown.

Analyze the modified equation

The question asks about the equation \(6 = -16x^2 + 24x + 1\).
In this equation, the height variable \(y\) has been replaced by the constant value \(6\).
Therefore, this equation represents the situation when the height of the football is exactly \(6\text{ ft}\).

Interpret the meaning of the zero

A "zero" or solution to the equation \(6 = -16x^2 + 24x + 1\) is a value of \(x\) (time in seconds) that makes the equation true.
Thus, the zero represents the time in seconds when the ball is 6 ft above the ground.

Evaluate the given options

• "the height of the ball after 6 seconds": Incorrect, as \(6\) is the height \(y\), not the time \(x\).
• "when the ball is 6 ft above the ground": Correct, as solving for \(x\) gives the time when the height \(y = 6\).
• "the initial upward velocity of the ball": Incorrect, the initial velocity is represented by the coefficient of the linear term, which is \(24\text{ ft/s}\).
• "the horizontal distance traveled by the ball when it is 6 ft above the ground": Incorrect, \(x\) represents time, not horizontal distance.

Answer:

• the height of the ball after 6 seconds
• when the ball is 6 ft above the ground (Correct answer)
• the initial upward velocity of the ball
• the horizontal distance traveled by the ball when it is 6 ft above the ground