The motion of a ball thrown off a cliff with an initial upward velocity and then affected by gravity would best be modeled by a quadratic function.
In physics, the height of an object under the influence of gravity can be described by a parabolic equation of the form:
\[ h(t) = -\frac{1}{2}gt^2 + v_0t + h_0 \]
Where:
- \( h(t) \) is the height at time \( t \),
- \( g \) is the acceleration due to gravity (approximately \( 9.81 , \text{m/s}^2 \) downwards),
- \( v_0 \) is the initial upward velocity (5 m/s in this case),
- \( h_0 \) is the initial height from which the ball is thrown (the height of the cliff).
The negative coefficient of the \( t^2 \) term indicates that the height will reach a maximum at some point before it begins to decrease back towards the ground.
Linear functions would imply a constant rate of change, which does not apply here as the height is not changing at a constant rate due to the effect of gravity. Exponential functions are not suitable either, as they typically describe growth or decay that accelerates or decelerates multiplicatively rather than as a result of uniform acceleration such as gravity.
Thus, the correct choice for modeling the height of the ball over time is a quadratic function.