To determine the factors of the quadratic function that models the height of the pumpkin over time, we'll start by identifying the zeros (also known as roots or x-intercepts) of the function. The zeros are the points where the height of the pumpkin, [tex]\( h(t) \)[/tex], is zero.
According to the data provided:
- At [tex]\( t = 0 \)[/tex] seconds, the height [tex]\( h = 0 \)[/tex] feet.
- At [tex]\( t = 2 \)[/tex] seconds, the height [tex]\( h = 272 \)[/tex] feet.
- At [tex]\( t = 10.5 \)[/tex] seconds, the height [tex]\( h = 0 \)[/tex] feet.
From this data, we can observe that the height of the pumpkin is zero at [tex]\( t = 0 \)[/tex] seconds and [tex]\( t = 10.5 \)[/tex] seconds. Therefore, the zeros of the quadratic function are:
- [tex]\( t = 0 \)[/tex] seconds
- [tex]\( t = 10.5 \)[/tex] seconds
These zeros (roots) of the quadratic function indicate that the quadratic function can be factored in terms of [tex]\( t \)[/tex] as follows:
[tex]\[ h(t) = a(t - 0)(t - 10.5) \][/tex]
Where [tex]\( a \)[/tex] is a constant that will determine the specific shape and orientation of the parabola, but is not necessary to identify the zeros.
Thus, the two factors of the quadratic function, obtained by identifying the zeros in the data points, are [tex]\( t = 0 \)[/tex] and [tex]\( t = 10.5 \)[/tex].
