The given function is \( f(x) = -3x^2 + 1 \). This is a quadratic function, and it represents a parabola that opens downwards because the coefficient of \( x^2 \) is negative.

### Key Features of the Function:
1. **Vertex**: The vertex form of a quadratic function is \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex. In this case, the vertex occurs at \( x = 0 \), and the function simplifies to \( f(0) = -3(0)^2 + 1 = 1 \). Thus, the vertex is \( (0, 1) \).
2. **Axis of Symmetry**: The axis of symmetry is the vertical line that passes through the vertex, \( x = 0 \).
3. **Direction**: Since the coefficient of \( x^2 \) is negative, the parabola opens **downward**.
4. **Y-intercept**: The y-intercept occurs when \( x = 0 \), so \( f(0) = 1 \). Therefore, the y-intercept is \( (0, 1) \).
5. **X-intercepts**: To find the x-intercepts, set \( f(x) = 0 \) and solve for \( x \):
   \[
   -3x^2 + 1 = 0 \quad \Rightarrow \quad x^2 = \frac{1}{3} \quad \Rightarrow \quad x = \pm \sqrt{\frac{1}{3}} = \pm \frac{\sqrt{3}}{3}
   \]
   So, the x-intercepts are \( \left( \frac{\sqrt{3}}{3}, 0 \right) \) and \( \left( -\frac{\sqrt{3}}{3}, 0 \right) \).

Would you like more details on any of these features or have any questions?

Here are 5 related questions to expand on this:
1. How do you convert the quadratic function to vertex form?
2. Can you explain how to calculate the vertex of a parabola algebraically?
3. How would this graph change if the coefficient of \( x^2 \) was positive?
4. What is the range of the function \( f(x) = -3x^2 + 1 \)?
5. How do you calculate the domain of any quadratic function?

**Tip**: For any quadratic function, the domain is always all real numbers \( (-\infty, \infty) \), but the range depends on whether the parabola opens upwards or downwards.

Explore the quadratic function f(x) = -3x^2 + 1, including its key features such as the vertex, axis of symmetry, direction, and intercepts. Understand the properties of downward-opening parabolas and learn how to calculate the x- and y-intercepts. Suitable for students in Grades 9-12.

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