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.