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resuelve : la parabola tiene el vetice en (1/2,-2) y los cortes con el eje x son: -2 y 2

The image shows a parabola with a vertex at \(\left(\frac{1}{2}, -2\right)\) and intercepts with the x-axis at \(x = -2\) and \(x = 2\).

### Step 1: Write the equation of the parabola
Given the vertex and x-intercepts, we can express the equation of the parabola in vertex form and standard form.

1. **Vertex form** of a parabola:
\[
y = a(x - h)^2 + k
\]
Where \((h, k)\) is the vertex of the parabola.

Given:
- \(h = \frac{1}{2}\)
- \(k = -2\)

So, the equation becomes:
\[
y = a\left(x - \frac{1}{2}\right)^2 - 2
\]

2. **Using the x-intercepts**:
Since the parabola intercepts the x-axis at \(x = -2\) and \(x = 2\), we can write the equation in factored form:
\[
y = a(x + 2)(x - 2)
\]

### Step 2: Determine the value of \(a\)
To find \(a\), we can expand the factored form and equate it to the vertex form:

1. Expand the factored form:
\[
y = a(x^2 - 4)
\]
2. Using the vertex form:
\[
y = a\left(x - \frac{1}{2}\right)^2 - 2
\]

Since both equations represent the same parabola, we need to equate them at a specific point. Use the vertex point \(x = \frac{1}{2}\) and \(y = -2\) to find \(a\).

Plug \(x = \frac{1}{2}\) into \(y = a(x^2 - 4)\):
\[
-2 = a\left(\left(\frac{1}{2}\right)^2 - 4\right) 
\]
\[
-2 = a\left(\frac{1}{4} - 4\right)
\]
\[
-2 = a\left(\frac{1}{4} - \frac{16}{4}\right)
\]
\[
-2 = a\left(\frac{-15}{4}\right)
\]

Solve for \(a\):
\[
a = \frac{-2 \times 4}{-15} = \frac{8}{15}
\]

Thus, the equation of the parabola is:
\[
y = \frac{8}{15}(x^2 - 4)
\]

Or, more commonly, by simplifying the equation:
\[
y = \frac{8}{15}(x - \frac{1}{2})^2 - 2
\]

### Conclusion
The equation of the parabola with a vertex at \(\left(\frac{1}{2}, -2\right)\) and x-intercepts at \(x = -2\) and \(x = 2\) is:
\[
y = \frac{8}{15}(x^2 - 4)
\]

Feel free to ask for further details or additional clarification!

Here are some related questions you might consider:
1. How do you convert between vertex form and standard form of a parabola?
2. What is the significance of the coefficient \(a\) in a parabola's equation?
3. How does the vertex of a parabola determine the direction of its opening?
4. How can you determine the y-intercept of a parabola given its equation?
5. What role do the x-intercepts play in graphing a parabola?

**Tip:** Always double-check the coordinates of the vertex and intercepts when deriving the equation of a parabola to ensure accuracy.

Learn how to find the equation of a parabola given its vertex at (1/2, -2) and x-intercepts at -2 and 2. This problem involves applying the vertex form and factoring methods to derive the equation. Suitable for high school students studying quadratic equations.

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