can you solve this equation vertex (4,2) axis of symmetry x =4, length of latus rectum =1/3,a>0   ? using vertex form

To solve the equation based on the given information, let's break it down step by step:

1. **Vertex**: The vertex of the parabola is at the point \((4, 2)\).
2. **Axis of symmetry**: The axis of symmetry is \(x = 4\), which means the parabola either opens upwards or downwards.
3. **Length of latus rectum**: The length of the latus rectum is \( \frac{1}{3} \). The latus rectum is the distance between two points on the parabola that are directly across from the focus, perpendicular to the axis of symmetry. The length of the latus rectum is given by \(4p\), so \(4p = \frac{1}{3}\).
4. **\(a > 0\)**: Since \(a > 0\), the parabola opens upwards.

### Steps to Construct the Vertex Form:

1. **Calculate \(p\)**:
   \[
   4p = \frac{1}{3}
   \]
   \[
   p = \frac{1}{12}
   \]
   Here, \(p\) is the distance from the vertex to the focus.

2. **Vertex Form of the Equation**:
   The vertex form of a parabola that opens upwards or downwards is:
   \[
   (x - h)^2 = 4p(y - k)
   \]
   Given that the vertex is \((4, 2)\) and the parabola opens upwards (since \(a > 0\)), we have:
   \[
   (x - 4)^2 = 4p(y - 2)
   \]
   Substituting \(p = \frac{1}{12}\) into the equation:
   \[
   (x - 4)^2 = 4\left(\frac{1}{12}\right)(y - 2)
   \]
   \[
   (x - 4)^2 = \frac{1}{3}(y - 2)
   \]

3. **Final Equation**:
   The equation of the parabola in vertex form is:
   \[
   (x - 4)^2 = \frac{1}{3}(y - 2)
   \]

This represents a parabola with its vertex at \((4, 2)\), opening upwards, and a latus rectum length of \(\frac{1}{3}\).

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 downwards instead of upwards?
2. What is the directrix of the parabola based on the given information?
3. How would you determine the focus of this parabola?
4. Can you rewrite this equation in the standard form?
5. How does the length of the latus rectum affect the shape of the parabola?

**Tip:** The latus rectum gives insight into the "width" of the parabola at the focus, which is particularly useful when graphing or understanding the parabola's geometry.

Learn how to solve a parabola equation given its vertex at (4, 2), axis of symmetry x = 4, and length of latus rectum 1/3. Understand the vertex form of a parabola, including the calculation of latus rectum. Suitable for high school students.

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