You shoot a beam of electrons through a double slit to make an interference pattern. after noting the properties of the pattern, you then double the speed of the electrons. what effect would this have?

The distance between bright patterns y is dictated by the formula: 

y=(λ*D)/d, where λ is the de Broglie wavelength, D is the distance from the double slit to the screen and d is the spacing between the slits. 

Since de Broglie wavelength changes like:

λ=h/p, where h is the Planck's constant and p is the momentum.

And the momentum is p=m*v, where m is the mass of the electron and v is the velocity. 

Since we can see that λ=h/m*v so that the higher the frequency the smaller the wavelength, this is going to have an impact on the interference pattern. 

y={(h/m*v)*D}/d=(h*D)/(d*m*v), we see that as we double the velocity v, or in other words, we put 2*v that y is going to be 2 times smaller because:

y=(h*D)/(2*d*m*v), we have number 2 in the denominator. We can rewrite this as: 

y=(1/2)*{(h*D)/(d*m*v)}

So the effect of doubling the speed of the electron is going to shrink the distance between the bright patterns in half. 

Alinear relationship would look like a straight line on the graph. it could have a positive negative or 0 slope. so it could be a line potentially going in any direction but it must be perfectly straight.