https://file.aiquickdraw.com/mathbot/file/1728730313016spyeal3b.jpg

The equation shown in the image is:

\[
(x - 5)^2 = 8(y - 1)
\]

This is the equation of a parabola. Here's a breakdown of the form:

- This is a standard equation of a parabola, specifically of the form \((x - h)^2 = 4p(y - k)\), which opens either upwards or downwards depending on the sign of \(4p\).
- In this case, \(h = 5\), \(k = 1\), and \(4p = 8\), so \(p = 2\).
  
The vertex of the parabola is at \((h, k) = (5, 1)\), and since \(p = 2\), the parabola opens upwards (because \(p > 0\)) and is stretched by a factor of 8 along the y-axis.

Would you like more details on how to plot or solve this parabola for specific values of \(x\) or \(y\)?

Here are 5 related questions to explore further:
1. How do you determine the direction in which a parabola opens from its equation?
2. What is the focus and directrix of this particular parabola?
3. How do you convert the equation of a parabola into standard form?
4. What is the relationship between the coefficient of \(y\) and the width of the parabola?
5. How can you find the axis of symmetry for this parabola?

**Tip**: For a parabola in the form \((x - h)^2 = 4p(y - k)\), the vertex is always at \((h, k)\), and the distance between the vertex and the focus is equal to \(p\).

Learn how to solve and analyze the parabola equation (x - 5)^2 = 8(y - 1). This problem involves conic sections, specifically parabolas, and is suitable for students in Grades 9-12. It explains the vertex form of the equation and how to determine the direction and stretching of the parabola.

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