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.