Question
Find a formula for thg exponential functkon passing through Ihe polnts ( 2 [ ard (3,128)
Find a formula for thg exponential functkon passing through Ihe polnts ( 2 [ ard (3,128)
Answers
Find a formula for an exponential function passing through the two points. $$ (-2,6),(3,1) $$
Already we're gonna start with our general equation F. Of X equals A B. To the X. And now we're gonna have our to coordinate points negative one comma 3/2 and three comma 24. So I'm going to start with three halves equals a B to the -1. And I'm gonna solve for A. So this is really 3/2 equals a over B. Multiply both sides by B. And we get A equals three B over to. So I'm going to put F. Of X equals three b. over to times B to the X. So three B over to just replace the a. Now I will substitute in my next point. So I'm gonna say three, nope 24 equals three B. Over to B. To the third. So I'm going to multiply both sides by two And I'm getting 48 equals three B. Times B to the third is three B to the 4th, divide by three. And I get B to the 4th equals 16. And now I want to take the 4th root of both sides. So the fourth root of 16 is too, so be equals two. So you can either take the 4th root Or you can take 16 to the 1 4th power, And you get B equals two. So now I substitute too Back in here and I get three times 2 divided by two A equals three. And now putting it all together, I get F of X equals three Times 2 to The X. Power.
We're going to start with our formula F. Of X equals a B to the X. And we have our two points negative 34 and 32. So I'm gonna plug in This point 1st. So I'm going to have to people's A B to the third, solving for a, dividing both sides. By being cute. I get to Over B to the 3rd. Now I use my other point so I'm going to have half of X equals two over B to the 3rd times B to the X. So I have four equals To over B to the 3rd times B To the negative three negative three, puts it in the denominator. So I have four equals two over BQ Times one over B. Cube. Which means for Equals two over B to the six, Which means I multiply both sides by B to the 6th. And I get four B to the 6th equals two, Divide both sides by four. So you can be to the sick Equals 1/2 which is .5. And then I take the the 6th route or to the 16 power. And that gives me be To the 1 6th power and has to in parentheses And be with equal .891. So I go back here to a and I put .891 in. So I get two divided by .891 Raised to the 3rd, and I get 2.827. For a. Now I put it all together and I get F of X Equals 2.8 27 times .89 one raised to the X.
Finding an exponential function That passes through zero, and two comma 20. Using half of X equals a. B to the X. I'm going to start with F F zero which equals a B to the zero, which equals 2000. So b equals one can be to the zero power is one, so I am a equals 2000. Now we're going to use the other point. So I have F M X equals 2000 B. To the X. Of two equals 2000 B. To the second equals 20. So now I have 2000 B to the second equals 20 Divide both sides by 2000. And I get one over 100 B squared Equals one over 100. That means I take the square root of both sides and I get B Equalling 1/10. So my the equation is F. Of X equals 2000 B. is 1 10th to the X. Power.
Starting with F. Of X. People's A. B. To the X. Were given two points two comma five six comma nine. So I Going to use my 1.5 equals a. B squared, divide both sides by B squared. A equals five over B squared. So now I go to my F. Of X. Substituting five over B squared for A. Times B to the X. Using my other point I put lugging A nine for F. Of X equals five over B squared times the to the sixth. So this ability over one. So I get nine equals five B. To the sixth over B squared. Subtract my exponents. And I get uh nine equals five. B. to the 4th, Divide both sides by B. to the 4th. I'm sorry, divide both sides by five. So we now have be to the 4th Equals 9/5. Take that to the one worth power and yeah. Mhm. I get right. nine divided by five raised to the 1/4 power parentheses. Mhm. Math enter, enter. No, but does not like that. All right. So b equals 1.158. So now I go back here and A equals five over 1.158 squared. I have divided by one 158 Squared. gives me one nope three point 7 to 9. Putting that all together. I get F of X Equals 3.729 Times 1.158 to the X. Power.