https://file.aiquickdraw.com/mathbot/file/1726974488451w4b2s9xb.jpg

The function provided in the image is \( f(x) = 3x^2 \), which is a quadratic function.

Key characteristics of the graph of \( f(x) = 3x^2 \):
1. The graph is a parabola.
2. Since the coefficient of \( x^2 \) is positive, the parabola opens upwards.
3. The vertex of the parabola is at the origin (0, 0).
4. The coefficient 3 causes the parabola to be narrower compared to \( f(x) = x^2 \).

Now, let's analyze the given graph choices:
- Option A shows a downward-facing parabola, which is incorrect because the parabola should open upwards.
- Option B shows an upward-facing parabola with a correct shape and vertex at (0, 0). This looks like the correct graph.
- Option C shows a parabola opening downwards and translated upwards, which is incorrect.
- Option D shows a wider parabola, which is incorrect because the coefficient of 3 should make the graph narrower.

The correct answer is **Option B**.

Would you like further details on this? Here are five related questions to deepen your understanding:
1. How does changing the coefficient of \(x^2\) affect the shape of the parabola?
2. What are the steps to graph a quadratic function by hand?
3. What happens to the parabola if we add or subtract a constant to \(3x^2\)?
4. How would the graph change if the function was \( f(x) = -3x^2 \)?
5. Can you describe how to find the vertex and axis of symmetry for any quadratic function?

**Tip:** The larger the absolute value of the coefficient of \(x^2\), the narrower the parabola will be.

Learn how to graph the quadratic function f(x) = 3x^2, analyze its key characteristics, and understand how the coefficient affects the shape of the parabola. This guide is suitable for high school students in Grades 9-11 and covers the vertex, axis of symmetry, and the effect of the coefficient on graph shape.

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