To analyze the given equation [tex]\( h(t) = -6(t - 3.7)^2 + 82.14 \)[/tex] and explain the significance of the term [tex]\(-3.7\)[/tex] and the value [tex]\( t = 3.7 \)[/tex], let's break it down step-by-step:
### Comparing Equations
First, we compare the given equation [tex]\( h(t) = -6(t - 3.7)^2 + 82.14 \)[/tex] with the basic quadratic function [tex]\( f(t) = -6t^2 \)[/tex]. Notice that the given equation is in the vertex form of a parabola, [tex]\( y = a(x - h)^2 + k \)[/tex].
### Understanding the Transformations
1. Horizontal Shift ([tex]\(-3.7\)[/tex]):
The term [tex]\((t - 3.7)\)[/tex] inside the squared term represents a horizontal translation of the graph. Specifically, it indicates a shift to the right by 3.7 units. This means that the entire graph of the function [tex]\( f(t) = -6t^2 \)[/tex] is moved 3.7 units to the right to obtain the graph of [tex]\( h(t) \)[/tex].
2. Vertical Translation (+82.14):
The constant term [tex]\( +82.14 \)[/tex] outside the squared term represents a vertical shift of the graph. It shifts the graph upward by 82.14 units. This determines the maximum height that the rocket reaches.
### Interpretation in the Context of the Rocket's Launch
Given that the vertex form of a parabola [tex]\( y = a(x - h)^2 + k \)[/tex] has its vertex at [tex]\((h, k)\)[/tex]:
- Horizontal Shift Interpretation:
The horizontal shift by [tex]\( \text{-}3.7 \)[/tex] means that the original vertex of the parabola [tex]\( f(t) = -6t^2 \)[/tex], which was at [tex]\( t = 0 \)[/tex], is now at [tex]\( t = 3.7 \)[/tex]. This is the point in time when the rocket reaches its maximum height.
- Maximum Height (82.14 feet):
The vertex of the parabola [tex]\( h(t) = -6(t - 3.7)^2 + 82.14 \)[/tex] is at the point [tex]\( (3.7, 82.14) \)[/tex]. This tells us that at [tex]\( t = 3.7 \)[/tex] seconds, the rocket reaches its peak height of 82.14 feet.
### Final Explanation
- Effect of the Term -3.7:
The term [tex]\(-3.7\)[/tex] causes a horizontal shift of the graph of the rocket's path to the right by 3.7 units.
- Significance of [tex]\( t = 3.7 \)[/tex]:
In the context of the rocket's flight, [tex]\( t = 3.7 \)[/tex] seconds is the time at which the rocket reaches its maximum height. At this moment, the height of the rocket is 82.14 feet.
This concludes that the term [tex]\(-3.7\)[/tex] causes the rocket's graph to shift right, making [tex]\( t = 3.7 \)[/tex] the specific time when the rocket is at its highest point during the launch.
