 In this video, we provide the solution to question number three for practice exam four from math 1050 We have the exponential function f of x equals c times a to the x We don't know c or a yet, but we do know that this exponential passes through the points 0 comma 80 and 2 comma 20 so using these two points. We're going to fill in for c and a right here That's what you have to figure out now. Notice that the first point is given to you as the wider set This is the initial value of the exponential function if you have the wider set for an exponential That's what you want to plug in first because notice that if you plug in f of 0 Well, since it passes through the point 080 we know f of 0 equals 80 But we also know that c times a to the 0 even though we don't know what a is yet We do know that a to the 0 is equal to 1 so this is just going to give us the value c So so for this exponential function the initial value 80 is the coefficient c So now we know that f of x has the form 80 times a to the x that's where the second point is going to come into play here We can plug in 2 so f of 2 we know this is equal to 20 since it passes through the point 2 comma 20 But we also know this is going to equal 80 times a squared and so we're going to solve for a in this situation So the next thing to do is divide both sides of the equation by 80. We're trying to solve for a in this expression Of course 20 over 80 is the same thing as 2 over 8 and 2 goes into 8 4 4 times So we end up with 1 4th is equal to a squared We're then going to take the square root of both sides to get rid of the square We only have to take care of the positive square root Because if we took the negative square root that would imply a is negative and our assumptions about exponential functions Is that the x the exponential base has to always be positive? So this would give us that a equals one half so notice what we have here We have that a equals one half. We have c equals 80 So we have to look for a function that's going to look like f of x equals 80 Times one half to the x in which case then we select choice B as that's exactly what we just found