The equation \( y = -2x^2 + x \) represents a quadratic function. This is a downward-facing parabola because the coefficient of \( x^2 \) is negative.

### Key Features:
1. **Vertex**: The vertex form of a quadratic equation is \( y = a(x-h)^2 + k \), where \( (h, k) \) is the vertex. For the given equation, we can use the vertex formula:
   \[
   h = -\frac{b}{2a}, \quad k = f(h)
   \]
   Here, \( a = -2 \) and \( b = 1 \). Substituting these values:
   \[
   h = -\frac{1}{2(-2)} = \frac{1}{4}
   \]
   Now substitute \( h = \frac{1}{4} \) back into the equation to find \( k \):
   \[
   k = -2\left(\frac{1}{4}\right)^2 + \frac{1}{4} = -2\left(\frac{1}{16}\right) + \frac{1}{4} = -\frac{2}{16} + \frac{1}{4} = -\frac{1}{8} + \frac{2}{8} = \frac{1}{8}
   \]
   Thus, the vertex is \( \left( \frac{1}{4}, \frac{1}{8} \right) \).

2. **Axis of Symmetry**: The axis of symmetry is the vertical line that passes through the vertex, given by:
   \[
   x = \frac{1}{4}
   \]

3. **Y-Intercept**: The y-intercept occurs when \( x = 0 \):
   \[
   y = -2(0)^2 + 0 = 0
   \]
   So, the y-intercept is \( (0, 0) \).

4. **Direction**: Since the coefficient of \( x^2 \) is negative, the parabola opens downward.

5. **X-Intercepts**: To find the x-intercepts, set \( y = 0 \) and solve for \( x \):
   \[
   0 = -2x^2 + x
   \]
   Factor the equation:
   \[
   0 = x(-2x + 1)
   \]
   So, \( x = 0 \) or \( -2x + 1 = 0 \). Solving for \( x \):
   \[
   x = \frac{1}{2}
   \]
   Therefore, the x-intercepts are \( (0, 0) \) and \( \left( \frac{1}{2}, 0 \right) \).

Would you like to go deeper into any part of this explanation?

### Related Questions:
1. How do we complete the square to transform a quadratic equation into vertex form?
2. What is the significance of the axis of symmetry in a quadratic function?
3. How do you find the focus and directrix of a parabola?
4. How can we solve quadratic equations using the quadratic formula?
5. What are the applications of quadratic functions in real-world problems?

**Tip**: When analyzing parabolas, always check the sign of the \( x^2 \)-term to determine the direction of the curve.

Explore the step-by-step solution for the quadratic equation y = -2x^2 + x. Learn about vertex calculation, axis of symmetry, and x-intercepts. This problem covers advanced algebraic concepts and is suitable for students in Grades 9-12.

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