To rewrite the quadratic function [tex]\( f(t) = 4t^2 - 8t + 6 \)[/tex] into its vertex form [tex]\( a(t-h)^2 + k \)[/tex], we will use the method of completing the square. Let's go through the steps in detail:
1. Identify coefficients:
- [tex]\( a = 4 \)[/tex]
- [tex]\( b = -8 \)[/tex]
- [tex]\( c = 6 \)[/tex]
2. Factor out the coefficient of [tex]\( t^2 \)[/tex] from the quadratic and linear terms:
[tex]\[
f(t) = 4(t^2 - 2t) + 6
\][/tex]
3. Complete the square inside the parentheses:
- Take the coefficient of [tex]\( t \)[/tex], which is [tex]\(-2\)[/tex], halve it to get [tex]\(-1\)[/tex], and then square it to get [tex]\(1\)[/tex].
- Add and subtract this square inside the parentheses.
[tex]\[
f(t) = 4(t^2 - 2t + 1 - 1) + 6
\][/tex]
[tex]\[
f(t) = 4((t-1)^2 - 1) + 6
\][/tex]
4. Simplify the expression:
- Distribute the 4:
[tex]\[
f(t) = 4(t-1)^2 - 4 + 6
\][/tex]
- Combine like terms:
[tex]\[
f(t) = 4(t-1)^2 + 2
\][/tex]
Now the function is in vertex form [tex]\( f(t) = a(t-h)^2 + k \)[/tex], where:
- [tex]\( a = 4 \)[/tex]
- [tex]\( h = 1 \)[/tex]
- [tex]\( k = 2 \)[/tex]
The vertex form of [tex]\( f(t) \)[/tex] is [tex]\( f(t) = 4(t-1)^2 + 2 \)[/tex].
### Interpretation of the Vertex
- The vertex of the parabola is at [tex]\( (h, k) = (1, 2) \)[/tex].
- Since [tex]\( a = 4 \)[/tex], which is positive, the parabola opens upwards and the vertex represents the minimum point.
- Therefore, the minimum height of the roller coaster is [tex]\( k = 2 \)[/tex] meters from the ground.
Hence, the correct option is:
[tex]\[ f(t)=4(t-1)^2+2; \text{ the minimum height of the roller coaster is } 2 \text{ meters from the ground} \][/tex]
