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Let's solve the given problem.

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### Problem:  
We are asked to **find the equation of a parabola** with the following properties:
- **Vertex**: \( (0, 0) \)
- **Axis along the x-axis**
- **Passes through the point** \( (2, -1) \)

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### Solution:

Since the axis of the parabola is along the **x-axis**, its equation will be in the form:

\[
y^2 = 4ax
\]

where:
- \(a\) is a constant that determines how wide or narrow the parabola is.
- The vertex is at the origin \( (0, 0) \), and the axis of symmetry is the **x-axis**.

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### Step 1: Use the given point \( (2, -1) \)

We are given that the parabola passes through \( (2, -1) \). This means the point must satisfy the equation \( y^2 = 4ax \). 

- For \(x = 2\) and \(y = -1\), substitute these values into the equation:

\[
(-1)^2 = 4a(2)
\]

\[
1 = 8a
\]

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### Step 2: Solve for \(a\)

\[
a = \frac{1}{8}
\]

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### Step 3: Write the Equation of the Parabola

Now that we know \( a = \frac{1}{8} \), the equation of the parabola is:

\[
y^2 = 4 \times \frac{1}{8} \times x
\]

\[
y^2 = \frac{1}{2}x
\]

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### Final Equation:

\[
y^2 = \frac{1}{2}x
\]

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### Sketch (Description):

- **Vertex**: The vertex is at \( (0, 0) \).
- **Orientation**: Since the equation is \( y^2 = \frac{1}{2}x \), the parabola opens **rightward** along the positive x-axis.
- **Point on the parabola**: The parabola passes through the point \( (2, -1) \).

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Do you need a plot of the parabola or further clarifications?  

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Here are 5 related questions you could explore:

1. What happens if the axis of the parabola is along the y-axis instead?
2. How do you derive the general form of a parabola given any vertex?
3. How would the equation change if the parabola were shifted to another vertex, say \( (h, k) \)?
4. What are the focus and directrix of the parabola with this equation?
5. How would you convert this equation into parametric form?

**Tip:** Always substitute given points into your equation to verify solutions. This ensures accuracy when deriving parameters.

Sketch and find the equation of the parabola having vertex (0, 0), axis along the x-axis, and passing through (2, -1).

Learn how to find the equation of a parabola with its vertex at the origin, axis along the x-axis, and passing through a given point. This problem covers concepts from conic sections and algebra, suitable for students in Grades 10-12.

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