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Choice B is correct. Since the line of symmetry for the graph of a quadratic function contains the vertex of the graph, the x-coordinate of the vertex of the graph of $y=f\left(x\right)$ is the x-coordinate of the midpoint of its two x-intercepts. The midpoint of two points with x-coordinates $x_1$ and $x_2$ has x-coordinate $x_m$, where $x_m=\frac{x_1+x_2}{2}$. It′s given that the vertex is $\left(-3,6\right)$ and one of the x-intercepts is $\left(-\frac{17}{4},0\right)$. Substituting $-3$ for $x_m$ and $-\frac{17}{4}$ for $x_1$ in the equation $x_m=\frac{x_1+x_2}{2}$ yields $-3=\frac{-{\displaystyle \frac{17}{4}}+x_2}{2}$. Multiplying each side of this equation by $2$ yields $-6=-\frac{17}{4}+x_2$. Adding $\frac{17}{4}$ to each side of this equation yields $-\frac{7}{4}=x_2$. Therefore, the other x-intercept is $\left(-\frac{7}{4},0\right)$.  

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.