The equation provided is $y = 3x^2 + bx + 5$, where $b$ is a constant. The graph of this equation shows a parabola opening upwards, with the vertex being the lowest point on the graph.

Here’s the step-by-step analysis:

1. Vertex of a Parabola: The general form of a quadratic function is $y = ax^2 + bx + c$, where:

   • $a = 3$ (positive, so the parabola opens upwards).
   • $b$ is unknown, and $c = 5$.

2. Vertex Formula: The x-coordinate of the vertex of a parabola can be calculated using the formula: $x_{\text{vertex}} = -\frac{b}{2a}$ Since $a = 3$, we substitute: $x_{\text{vertex}} = -\frac{b}{2(3)} = -\frac{b}{6}$

3. Observation from the Graph: The graph has its vertex at the origin (0, 0). This means the x-coordinate of the vertex must be 0. Hence: $-\frac{b}{6} = 0$ Solving this gives: $b = 0$

Therefore, the value of $b$ is 0, which corresponds to option B.

Would you like further clarification or details on this solution? Here are a few related questions you might find useful:

1. How do you determine the vertex of a parabola from its equation?
2. What does the sign of the coefficient $a$ in a quadratic equation tell you about the direction of the parabola?
3. How can you find the value of $b$ if the vertex is not at the origin?
4. What would happen to the graph if $b$ were positive or negative?
5. Can you find the roots of this quadratic equation if you know the value of $b$?

Tip: To identify the direction of a parabola from its equation, look at the sign of $a$. If $a > 0$, the parabola opens upwards, and if $a < 0$, it opens downwards.