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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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.