The equation given is:

\[
x = -y^2 + 5y - 8
\]

This is a quadratic equation in \( y \) with respect to \( x \). Here's how you can analyze it:

### General form:
The equation resembles the standard form of a quadratic equation:

\[
x = ay^2 + by + c
\]

Where:
- \( a = -1 \) (coefficient of \( y^2 \))
- \( b = 5 \) (coefficient of \( y \))
- \( c = -8 \) (constant term)

### Key points:

1. **Vertex**: 
   The vertex of a parabola given by \( x = ay^2 + by + c \) can be found by using the formula for the \( y \)-coordinate of the vertex:
   
   \[
   y_{\text{vertex}} = \frac{-b}{2a} = \frac{-5}{2(-1)} = \frac{5}{2} = 2.5
   \]
   
   Substitute this \( y \)-value back into the original equation to find the \( x \)-coordinate of the vertex:
   
   \[
   x = -(2.5)^2 + 5(2.5) - 8 = -6.25 + 12.5 - 8 = -1.75
   \]

   So, the vertex is at \( (x, y) = (-1.75, 2.5) \).

2. **Direction of the parabola**: 
   Since \( a = -1 \) (a negative value), the parabola opens downwards.

3. **Axis of symmetry**:
   The axis of symmetry is a vertical line passing through the vertex, so it is \( y = 2.5 \).

4. **Y-intercepts**: 
   You can find the \( y \)-intercepts by setting \( x = 0 \) and solving for \( y \):

   \[
   0 = -y^2 + 5y - 8 \quad \text{or} \quad y^2 - 5y + 8 = 0
   \]
   The discriminant of this quadratic equation is:

   \[
   \Delta = b^2 - 4ac = 5^2 - 4(1)(8) = 25 - 32 = -7
   \]

   Since the discriminant is negative, there are no real \( y \)-intercepts (i.e., the graph does not cross the x-axis).

5. **X-intercepts**:
   Solving for \( x \) directly in terms of \( y \), the quadratic form helps describe the behavior of the equation for different \( y \)-values, but no x-intercepts were provided to calculate explicitly.

---

Do you want more details on the calculation, or do you have any questions? 

Here are 5 follow-up questions to deepen your understanding:

1. How would the graph change if the coefficient of \( y^2 \) were positive?
2. Can you determine the maximum or minimum value of \( x \) for this equation?
3. How would you solve for \( y \) in terms of \( x \)?
4. What happens if you shift the equation vertically or horizontally?
5. How does the discriminant help determine the number of intercepts for a quadratic equation?

**Tip**: The vertex form of a quadratic equation can be helpful for identifying transformations and key points like the vertex more easily.

Explore the quadratic equation x = -y² + 5y - 8 with a step-by-step breakdown. Learn how to find the vertex, determine the parabola's direction, and analyze intercepts using key formulas such as the vertex and discriminant. Suitable for Grades 9-12 students.

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