The equation \( Y = 2x^2 + 3 \) represents a quadratic function. The general form of a quadratic function is \( Y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. In your equation:

- \( a = 2 \) (this is the coefficient of \( x^2 \))
- \( b = 0 \) (since there is no \( x \) term, \( b = 0 \))
- \( c = 3 \) (this is the constant term)

### Key Features of the Quadratic Function:
1. **Vertex**: The vertex of the parabola described by this function occurs at the point \( (h, k) \). Since \( b = 0 \), the x-coordinate of the vertex is \( x = 0 \). The y-coordinate is \( Y = c = 3 \). Therefore, the vertex is at \( (0, 3) \).
  
2. **Axis of Symmetry**: The axis of symmetry is the vertical line that passes through the vertex. Here, it's the line \( x = 0 \).

3. **Direction of Opening**: Since \( a = 2 \) (positive), the parabola opens upwards.

4. **Y-intercept**: The y-intercept is the value of \( Y \) when \( x = 0 \). Substituting \( x = 0 \) into the equation gives \( Y = 3 \). Thus, the y-intercept is \( (0, 3) \).

5. **Shape of the Parabola**: The parabola will be narrower than the standard parabola \( Y = x^2 \) because the coefficient \( a = 2 \) is greater than 1.

Would you like more details or have any questions?

Here are 5 questions that expand on this information:
1. How do you find the roots of the equation \( Y = 2x^2 + 3 \)?
2. What would happen to the graph if the coefficient \( a \) was negative?
3. How would you calculate the point where the parabola intersects the x-axis?
4. How does the value of \( c \) affect the position of the parabola?
5. Can you explain how to find the vertex using the formula for the axis of symmetry?

**Tip**: To graph quadratic functions easily, focus on the vertex, axis of symmetry, and the direction the parabola opens.

Explore how to analyze and graph the quadratic function Y=2x²+3. This tutorial covers key algebraic concepts such as identifying the vertex, axis of symmetry, and direction of the parabola. Suitable for students in Grades 8-10.

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