can you solve this equation vertex at origin, open left, 1/8 units between the vertex and focus  ? using vertex form

To solve this equation, let's interpret the given information step by step:

1. **Vertex at origin**: The vertex of the parabola is at the point \((0, 0)\).
2. **Opens left**: The parabola opens to the left, which implies that the equation will involve \((y - k)^2 = -4p(x - h)\) in vertex form, where \((h, k)\) is the vertex.
3. **1/8 units between the vertex and focus**: The distance between the vertex and the focus is \(p\), and it is given as \(p = \frac{1}{8}\).

Now let's construct the vertex form of the equation:

### Vertex Form:
The general vertex form for a parabola that opens left (or right) is:
\[
(y - k)^2 = -4p(x - h)
\]

Given that the vertex is at the origin \((0, 0)\) and the parabola opens to the left, the equation simplifies to:
\[
y^2 = -4p \cdot x
\]

### Plugging in the Distance \(p\):
Since \(p = \frac{1}{8}\), we substitute this value into the equation:
\[
y^2 = -4\left(\frac{1}{8}\right)x
\]
\[
y^2 = -\frac{1}{2}x
\]

### Final Equation:
The equation of the parabola in vertex form is:
\[
y^2 = -\frac{1}{2}x
\]

This represents a parabola that opens to the left, with its vertex at the origin and a focus located \(\frac{1}{8}\) units to the left of the vertex.

Would you like more details or have any questions?

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Here are some related questions to deepen your understanding:

1. How would the equation change if the parabola opened to the right instead of the left?
2. What is the equation of a parabola that opens upwards with the same vertex and focus distance?
3. Can you derive the equation of a parabola given a different vertex position?
4. How would you find the directrix of the parabola given in this problem?
5. What is the standard form of a parabola and how does it relate to the vertex form?

**Tip:** Remember that the sign of the \(4p\) term in the vertex form of a parabola's equation determines the direction in which the parabola opens.

Learn how to solve a parabola equation with vertex at origin and opening left. Step-by-step explanation using vertex form and distance formulas. Suitable for high school students.

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