How can I write an exponential function whose graph passes through the points (0, -6) and (-2,-54)?

You are looking for an equation of this form:  y  =  a·b^x

Using the point (0, -6)   --->   -6  =  a·b⁰   --->   -6  =  a·1   --->   -6  =  a

So, now the equation is:  y  =  -6·b^x 

Now, use the point (-2, -54)   --->   -54  =  -6b^-2   --->   9  =  b^-2   --->   1/9  =  b²   --->   b  =  ±(1/3)

So:  either:  y  =  -6·(1/3)^x   or   y  =  -6·(-1/3)^x