https://r2.erweima.ai/mathbot/image/1725758504231ptsfdpgj.jpg

To solve this problem, we need to determine the positions of the moving object (móvel) at the given times \( t = 2, 3, \text{ and } -4 \) based on the quadratic function shown in the graph.

From the graph, the function appears to represent a parabola. The key points on the parabola are at:
- The vertex is at \( (0, 16) \), which suggests the function might be of the form \( g(t) = -at^2 + 16 \), where \( a \) is a constant.
- The parabola is symmetric around the y-axis, confirming that it is in the form \( g(t) = -at^2 + 16 \).

### Step-by-Step Solution

1. **Determine the Function Form:**

   From the graph, we see the parabola is opening downwards and has a vertex at \( (0, 16) \). This means the quadratic function is likely:

   \[
   g(t) = -at^2 + 16
   \]

   To find \( a \), we use another point on the graph, such as \( t = 4 \) where \( g(t) = 0 \).

   \[
   0 = -a(4)^2 + 16 \implies 0 = -16a + 16 \implies a = 1
   \]

   So, the function is:

   \[
   g(t) = -t^2 + 16
   \]

2. **Find the Positions at Given Times:**

   - For \( t = 2 \):

     \[
     g(2) = -(2)^2 + 16 = -4 + 16 = 12
     \]

   - For \( t = 3 \):

     \[
     g(3) = -(3)^2 + 16 = -9 + 16 = 7
     \]

   - For \( t = -4 \):

     \[
     g(-4) = -(-4)^2 + 16 = -16 + 16 = 0
     \]

### Final Answers

- \( g(2) = 12 \)
- \( g(3) = 7 \)
- \( g(-4) = 0 \)

These are the positions occupied by the moving object at the respective times.

Would you like further details or explanations? Here are 5 related questions for deeper understanding:

1. What does the vertex of a quadratic function represent in a real-world scenario?
2. How do we determine the direction in which a parabola opens?
3. What is the significance of the coefficient \( a \) in a quadratic function \( ax^2 + bx + c \)?
4. How can we find the axis of symmetry of a quadratic function?
5. Can you explain how to find the roots of a quadratic function and their relevance?

**Tip:** The coefficient of \( t^2 \) determines whether the parabola opens upwards or downwards.

Quais são as posições ocupadas pelo móvel nos seguintes instantes: t = 2, t = 3, t = -4?

Learn how to determine the positions occupied by a moving object at specific times (t = 2, t = 3, t = -4) using a quadratic function g(t) = -t^2 + 16. This problem involves concepts of quadratic functions, algebra, and parabola symmetry, and is suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation