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Choice A is correct. An equation for the graph shown can be written in slope-intercept form $y=mx+b$, where $m$ is the slope of the graph and its y-intercept is $\left(0,b\right)$. Since the y-intercept of the graph shown is $\left(0,2\right)$, the value of $b$ is $2$. Since the graph also passes through the point $\left(4,1\right)$, the slope can be calculated as $\frac{1-2}{4-0}$, or $-\frac{1}{4}$. Therefore, the value of $m$ is $-\frac{1}{4}$. Substituting $-\frac{1}{4}$ for $m$ and $2$ for $b$ in the equation $y=mx+b$ yields $y=-\frac{1}{4}x+2$. It’s given that an equation for the graph shown is $y=f\left(x\right)+14$. Substituting $f\left(x\right)+14$ for $y$ in the equation $y=-\frac{1}{4}x+2$  yields $f\left(x\right)+14=-\frac{1}{4}x+2$. Subtracting $14$ from both sides of this equation yields $f\left(x\right)=-\frac{1}{4}x-12$.

Choice B 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.