Step 1 :The given points $(-12,0)$ and $(2,0)$ are the x-intercepts of the sinusoidal function. The distance between these two points is $2 - (-12) = 14$ units on the x-axis.
Step 2 :The period of a sinusoidal function is the distance between two consecutive x-intercepts. Therefore, the period of the given function is $14$ units.
Step 3 :The standard form of a sine function is $f(x) = A\sin(B(x - C)) + D$, where $A$ is the amplitude, $B$ is the frequency, $C$ is the phase shift, and $D$ is the vertical shift.
Step 4 :Since the amplitude is not given, we assume it to be $1$. The frequency is the reciprocal of the period, so $B = \frac{1}{14}$.
Step 5 :The phase shift is the largest stated x-intercept, which is $2$. Therefore, $C = 2$.
Step 6 :Since the graph passes through the origin, the vertical shift $D = 0$.
Step 7 :Substituting these values into the standard form, we get $f(x) = \sin\left(\frac{1}{14}(x - 2)\right)$.
Step 8 :Therefore, the sinusoidal function that uses the largest stated x-intercept value as its phase shift and matches the given graph is $f(x) = \sin\left(\frac{1}{14}(x - 2)\right)$.
Step 9 :\boxed{f(x) = \sin\left(\frac{1}{14}(x - 2)\right)}