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all stars need to be collected using parabola equations

To collect all the stars using parabolas, we need to identify the approximate coordinates of the stars and create equations of parabolas that pass through or near these points. Here’s how we can approach it:

### Coordinates of the stars:
- The first star appears to be at \( (0, 4) \).
- The next two stars are at \( (2, 4) \) and \( (4, 4) \).
- The last star is at \( (4, 2) \).

We can create a parabola with the general form:

\[
y = a(x - h)^2 + k
\]

where:
- \( (h, k) \) is the vertex of the parabola.
- \( a \) controls the "width" and direction (upward or downward) of the parabola.

### First possible parabola
To collect the stars along \( y = 4 \), we can use a parabola that passes through points \( (0, 4) \), \( (2, 4) \), and \( (4, 4) \). Let's try the equation:

\[
y = -\frac{1}{2} (x - 2)^2 + 4
\]

This is a downward-facing parabola with vertex \( (2, 4) \), collecting the stars on the same horizontal line.

### Second parabola
We can create another parabola to collect the star at \( (4, 2) \). One way to do this is by shifting the vertex down. A possible equation could be:

\[
y = (x - 4)^2 + 2
\]

This is an upward-facing parabola with vertex \( (4, 2) \), which will collect the lower star.

By entering these two equations into the system, the parabola curves should pass through or near the stars, allowing them to be "collected."

Would you like to see the steps in further detail or have any questions?

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Here are 5 related questions to expand on this:

1. How can we adjust the vertex form of a parabola for other sets of points?
2. What happens when we change the \(a\) value in a parabola’s equation?
3. Can multiple parabolas intersect at the same point?
4. How do we calculate the focus and directrix of a parabola?
5. How do transformations like shifting and stretching affect a parabola’s graph?

**Tip:** When adjusting a parabola's equation, small changes to the \(a\), \(h\), or \(k\) values can significantly impact the curve's shape and position.

All stars need to be collected using parabola equations

Learn how to collect all stars in the given challenge by creating parabola equations. Understand the application of the vertex form of quadratic equations and coordinate geometry. Suitable for students in Grades 9-12.

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