Why is the income effect zero for quasilinear utility functions? Suppose I have the utility function $$U(x,y) = \sqrt{x} + y$$
subject to budget constraint 
$$p_x x + p_y y = m$$
Then 
$$x_M =\frac{p_y^2}{4 p_x^2}$$ 
$$y_M = \frac{m}{p_y} - \frac{p_y}{4 p_x}$$

where $M$ denotes Marshallian. 

Now suppose I increase $p_x$ to $p_x'$. 

Why is the income effect zero?
From the formula for $x_M$, we see it has no dependence on income $m$. So $$\frac{\partial x_M}{\partial m} =0$$ Thus, the Slutsky equation 

$$\frac{\partial x_M}{\partial p_x} = \frac{\partial x_H}{\partial p_x} +-\frac{\partial x_M}{\partial m}x_M$$

implies 

$$\frac{\partial x_M}{\partial p_x} = \frac{\partial x_H}{\partial p_x} +(0)x_M $$
$$\frac{\partial x_M}{\partial p_x} = \frac{\partial x_H}{\partial p_x}  $$
Hence, and since $-\frac{\partial x_M}{\partial m}x_M$ is the income effect, this implies the income effect is zero and all the change is due to the substitution effect.