Writing exponential functions from tables

consider the following table of values for a linear function f of X is equal to MX plus B and an exponential function G of X is equal to a times R to the X write the equation for each function and so they give us for each x-value what f of X is and what G of X is and we need to figure out the equation for each function and type them in over here so I copy and pasted this problem on my little scratch pad so let's first think about the linear function and to figure out a figure out the equation of a line or a linear function right over here you really just need two points and I always like to use the situation when x equals zero because that makes it very clear what the y-intercept is going to be so for example we can say that F of 0 F of 0 is going to be equal to M times 0 plus B or this is just going to be equal to B and they tell us that F of 0 is equal to 5 B is equal to 5 so we immediately know that this B right over here is equal to 5 now we just have to figure out the M we have to figure out the slope of this line so just as a little bit of a refresher on slope the slope of this line is going to be our change in Y or a change in our function I guess we could say if we say that this is y is equal to f of X over our change in X and actually let me write it that way we could write this as our change in our function over our change in X if you want to look at it that way so let's look at this first change in X when X goes from 0 to 1 so we finish it 1 we started at 0 and f of X finishes at 7 and started at 5 so when X is 1 f of X is 7 when x is 0 f of X is 5 and we get a change in our function of 2 when X changes by 1 so our M is equal to 2 and you see that when x increases by 1 our function increases by 2 so now we know the equation for f of X f of X is going to be equal to is going to be equal to 2 times 2x plus B or 5 so we figured out what f of X is now we need to figure out what G of X is so G of X is an exponential function and there's really two things that we need to figure out we need to figure out what a is and we need to figure out what R is and let me just rewrite that so we know that G of X maybe I'll do it down here G of X is equal to a times R to the X power and if we know what G of 0 is that's a pretty useful thing because R to the 0th power regardless of what R is where I guess we can forget assume that R it's not equal to zero that leads to people can debate what 0 to the 0 power is but if R is any nonzero number we know that you raise that to the 0 power you get 1 and so that essentially gives us a so let's just write that down G of 0 G of 0 is a times R to the 0 power which is just going to be equal to a times 1 or a and they tell us what G of 0 is G of 0 is equal to 3 so we know that a is equal to 3 so so far we know that our G of X can be written as 3 times R to the X power so now we can just use any one of the other values they gave us to solve for R for example they tell us that G of 1 is equal to 2 so let's write that down G of 1 which would be 3 times R to the first power or just 3 let me just write it could be 3 times R to the first power or we could just write that as 3 times R they tell us that G of 1 is equal to 2 is equal to 2 so we get 3 times R is equal to 2 or we get that R is equal to 2/3 divide both sides of this equation by 3 so R is 2/3 and we're done G of X G of X is equal to 3 times two thirds three times two thirds actually we just write it this way three times two thirds to the X power 3 times 2/3 to the X power you could write that way if you want any which way so 3 times 2/3 to the X power and f of X is 2x plus 5 so let's actually just type that in so f of X is 2x plus 5 f of X is 2x plus 5 and we can verify that that's the expression that we want and G of X is 3 times 2 over 3 2 over 3 to the to the X power let me just verify that that's what I did there have a short memory oh yeah that looks right all right let's check our answer and we got it right