The function f(t) = 4t2 − 8t + 7 shows the height from the ground f(t), in meters, of a roller coaster car at different times t. Write f(t) in the vertex form a(x − h)2 + k, where a, h, and k are integers, and interpret the vertex of f(t).

f(t) = 4(t − 1)2 + 3; the minimum height of the roller coaster is 1 meter from the ground
f(t) = 4(t − 1)2 + 3; the minimum height of the roller coaster is 3 meters from the ground
f(t) = 4(t − 1)2 + 2; the minimum height of the roller coaster is 2 meters from the ground
f(t) = 4(t − 1)2 + 2; the minimum height of the roller coaster is 1 meter from the ground

Let's start by rewriting the given quadratic function \( f(t) = 4t^2 - 8t + 7 \) into the vertex form \( f(t) = a(t - h)^2 + k \).

### Step 1: Identify the coefficients
The given function is:
\[
f(t) = 4t^2 - 8t + 7
\]
Here, \( a = 4 \).

### Step 2: Complete the square
We need to rewrite the quadratic part \( 4t^2 - 8t \) in a perfect square form.

1. Factor out the coefficient of \( t^2 \) from the first two terms:
   \[
   f(t) = 4(t^2 - 2t) + 7
   \]
2. To complete the square, take half of the coefficient of \( t \) in the parentheses (which is \(-2\)), square it, and add and subtract this square inside the parentheses.
   \[
   f(t) = 4\left(t^2 - 2t + 1 - 1\right) + 7
   \]
   \[
   f(t) = 4\left((t - 1)^2 - 1\right) + 7
   \]
3. Distribute the 4 and simplify:
   \[
   f(t) = 4(t - 1)^2 - 4 + 7
   \]
   \[
   f(t) = 4(t - 1)^2 + 3
   \]

### Step 3: Identify the vertex and interpret it
The function in vertex form is:
\[
f(t) = 4(t - 1)^2 + 3
\]
This tells us that the vertex of the parabola is at \( (h, k) = (1, 3) \).

### Interpretation:
- The vertex of the function \( f(t) = 4(t - 1)^2 + 3 \) is \( (1, 3) \).
- This means that the minimum height of the roller coaster car occurs at \( t = 1 \) second, and the minimum height is \( 3 \) meters from the ground.

### Correct Answer:
The correct interpretation is:
\[
\textbf{f(t) = 4(t - 1)^2 + 3; the minimum height of the roller coaster is 3 meters from the ground.}
\]

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Would you like more details or have any questions?

### Related Questions:
1. How do you find the vertex of a quadratic function given in standard form?
2. What is the significance of the coefficient "a" in the vertex form of a quadratic function?
3. How does the value of "h" in the vertex form relate to the graph of the quadratic function?
4. Can a quadratic function have a maximum value instead of a minimum? If so, when?
5. What does the vertex represent in the context of real-world problems, such as projectile motion?
6. How do you determine the axis of symmetry for a quadratic function?
7. What happens to the graph of a quadratic function if "a" is negative?
8. How do you convert a quadratic function from vertex form back to standard form?

### Tip:
Completing the square is a powerful method not just for finding the vertex form, but also for solving quadratic equations and integrating certain types of functions.

Learn how to convert a quadratic function to vertex form and interpret the minimum height of a roller coaster car using a detailed example. This problem involves quadratic functions, vertex form, and completing the square method. Suitable for high school students (Grades 9-12).

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