The equation given is a quadratic function:

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

This equation represents a parabola. Here's a breakdown of its components:

1. **Quadratic Term ( \(2x^2\) )**: This term determines the shape and direction of the parabola. Since the coefficient of \(x^2\) is positive (2), the parabola opens upwards.

2. **Linear Term ( \(x\) )**: This term affects the slope and the tilt of the parabola.

3. **Constant Term (1)**: This term shifts the parabola up or down on the y-axis.

### Key Features of the Parabola:

- **Vertex**: The vertex is the highest or lowest point of the parabola. The vertex can be found using the formula \(x = -\frac{b}{2a}\). For this equation, \(a = 2\), \(b = 1\), and \(c = 1\). 

- **Axis of Symmetry**: The line of symmetry of the parabola is \(x = -\frac{b}{2a}\).

- **Y-intercept**: The y-intercept occurs where \(x = 0\). Substituting \(x = 0\) in the equation gives \(y = 1\).

### Do you want to calculate the vertex, axis of symmetry, and other properties in detail? 

Here are five questions that build on this topic:

1. How do you calculate the vertex of a quadratic equation?
2. What is the significance of the axis of symmetry in a parabola?
3. How does the value of the coefficient \(a\) affect the width of the parabola?
4. How do you find the x-intercepts of a quadratic equation?
5. What are some real-life applications of quadratic equations?

**Tip**: Always check the sign of the leading coefficient (\(a\)) to determine the direction in which a parabola opens (upward if positive, downward if negative).

Explore the quadratic equation y = 2x^2 + x + 1 in depth, including its components such as the vertex, axis of symmetry, and y-intercept. Learn how to find these properties and understand the behavior of the parabola. Suitable for Grades 10-12 students.

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