Is it possible to take the transfer function of a transistor?
Biased like that, the transistor is in saturation, and will not amplify.

Set r2 = r/2, and you'll have a large and nearly linear output range, with gain of -1.

Now add a series RC in parallel with the emitter resistor, and make that new resistor be r/10; you'll have a rather linear gain = 10x.

Now for some values:

R1 = 100K ohms (brown/black/yellow)

R2 = 47K ohms (yellow/purple/orange)

R3 = 5.1K ohms (green/brown/red)

new resistor in series RC: R_gain_set = 510 ohms (green/brown/brown)
and that new capacitor ithe series RC C_low_corner = 22uf 
(and check the polarity) for 16 cycle per second 3dB rolloff at the low/bass.

R4 = 5.1K ohms

With base at VDD/3, and emitter a  little lower, we guarantee the collector will be a little higher than 2/3 VDD, and we are guaranteed the transistor Vce is about VDD/3 which is very nicely far from saturation.

life is good.

if beta >> 20, life is really good.

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lets discuss distortion. 

If the emitter is grounded, such that the input AC signal appears across base to emitter, then a 4milliVolt peak peak input (0.004 volts) causes about 10% distortion.

In our added RC series network across the R3 resistor, if the R portion is ZERO, having only a capacitor in parallel with R3, you've in effect AC_Grounded that emitter, and 4mVPP input will cause 10% distortion.

However the distortion of the base-emitter junction is predicted by a polynomial that gives the designer the coefficients for each order of distortion. 

Thus a polynomial like


Vout = K0 + K1 * Vin + K2 * (Vin * Vin) + K3 * (Vin * Vin * Vin) + ....


can be very useful to people evaluating low distortion audio circuits or low distortion ADC preamplifiers or low distortion Radio Frequency amplifiers, or high distortion guitar amplifiers.

The 2nd order term predicts A * B intermodulation distortion, as well as predicting A * A (self) distortion. Examine the 2nd Order Intercept as useful.

The 3rd order term predicts A * A * A (self) distortion, as well as predicting A * A * B and A * B * B (which you will find are symmetric effects).
Examine the 3rd Order Intercept as useful.

NOW THE KEY IDEA: if you have a resistor in series with the emitter, and there is NO CAPACITOR across that resistor, you can dial up the desired amount of distortion. 

A resistor with DC drop of 10 * 0.026 volts, will greatly reduce the distortion.

A resistor with DC drop of 100 * 0.026 volts, produces even less distortion.

You can re-derive the polynomial, with a linear resistor added onto the exponential base-emitter diode non-linearity, and have the revised math for this.