https://r2.erweima.ai/mathbot/image/1726789123359vpxgg1fg.jpg

The graph in the image shows a parabolic trajectory, and the equation of motion follows the form of a quadratic function. From the graph, we can interpret that it represents the vertical position \( y(t) \) of an object moving over time \( t \).

The key points visible in the graph are:

- The vertex of the parabola is at \( t = 0 \), with the maximum height \( y = 8 \, m \).
- The parabola crosses the time axis at two points: \( t = -4 \) and \( t = 4 \).
  
This suggests the equation for \( y(t) \) could follow the form:
\[
y(t) = a(t - h)^2 + k
\]
where \( (h, k) \) is the vertex of the parabola, meaning \( h = 0 \) and \( k = 8 \).

Given that the parabola is symmetric and crosses the vertical axis at \( t = -4 \) and \( t = 4 \), we can solve for the coefficient \( a \) using the fact that at \( t = 4 \), \( y = 0 \).

\[
0 = a(4 - 0)^2 + 8
\]
\[
0 = 16a + 8
\]
\[
a = -\frac{1}{2}
\]

Thus, the equation of motion is:
\[
y(t) = -\frac{1}{2}t^2 + 8
\]

Now, let's compute the values of \( y(t) \) at the given times \( t = 1, 0, -2, -4 \):

1. For \( t = 1 \):
\[
y(1) = -\frac{1}{2}(1)^2 + 8 = -\frac{1}{2} + 8 = 7.5
\]

2. For \( t = 0 \):
\[
y(0) = -\frac{1}{2}(0)^2 + 8 = 8
\]

3. For \( t = -2 \):
\[
y(-2) = -\frac{1}{2}(-2)^2 + 8 = -\frac{1}{2}(4) + 8 = -2 + 8 = 6
\]

4. For \( t = -4 \):
\[
y(-4) = -\frac{1}{2}(-4)^2 + 8 = -\frac{1}{2}(16) + 8 = -8 + 8 = 0
\]

### Final answers:
- \( t = 1 \), \( y = 7.5 \)
- \( t = 0 \), \( y = 8 \)
- \( t = -2 \), \( y = 6 \)
- \( t = -4 \), \( y = 0 \)

Would you like more details or clarifications?

Here are five additional questions to deepen your understanding:
1. How can you derive the general form of a quadratic function from its vertex?
2. What physical conditions determine the shape of a parabolic motion graph?
3. How do the coefficients of a quadratic equation affect the direction and width of the parabola?
4. What is the significance of the parabola's axis of symmetry in this context?
5. How would the equation change if the object reached a different maximum height?

**Tip**: Understanding the vertex form of a quadratic equation helps in quickly identifying the maximum or minimum points of motion!

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Learn how to determine the positions of an object at different time intervals using a quadratic equation. This math problem involves calculating the object's vertical displacement over time using the vertex form of a quadratic function. Suitable for high school students.

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