The equation you provided is a quadratic function:

\[ y = 2x^2 + x + 1 \]

### Key Characteristics:
1. **Type of Equation**: Quadratic (degree 2).
2. **Standard Form**: \( y = ax^2 + bx + c \), where \( a = 2 \), \( b = 1 \), and \( c = 1 \).
3. **Direction of Parabola**: Since \( a = 2 > 0 \), the parabola opens upwards.
4. **Vertex**: The vertex can be found using the formula \( x = -\frac{b}{2a} \).
   \[
   x = -\frac{1}{2(2)} = -\frac{1}{4}
   \]
   Plugging this back into the equation to find \( y \):
   \[
   y = 2\left(-\frac{1}{4}\right)^2 + \left(-\frac{1}{4}\right) + 1 = \frac{1}{8} - \frac{1}{4} + 1 = \frac{7}{8}
   \]
   So, the vertex is at \( \left(-\frac{1}{4}, \frac{7}{8}\right) \).
5. **Y-intercept**: This occurs when \( x = 0 \), so \( y = 1 \).

### Do you want more details or have any questions?

#### Related Questions:
1. How do you find the roots of this quadratic equation?
2. What is the axis of symmetry for this function?
3. How do you determine if the function has a maximum or minimum value?
4. How do you graph this quadratic function?
5. What is the significance of the discriminant in this equation?

**Tip:** The discriminant (\( b^2 - 4ac \)) tells you the number and type of roots for a quadratic equation.

Explore the quadratic equation 2x^2 + x + 1 with a detailed explanation of its properties and solutions. Learn about the vertex, y-intercept, and direction of the parabola. Suitable for high school students studying algebra and quadratic functions.

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