In this problem, we are going to answer the following question. Given two points (1,1) and...

 

Transcribed Image Text:

In this problem, we are going to answer the following question. Given two points (1,1) and (2,2), can we find an exponential function y = c(a), such that it passes through both of these points? If so, how many such functions can we find? a. Let us for example, take an arbitrary two points (3, 16) and (2,8). If there is an expo- nential function y = c(a)" which passes through these two points then r = 3, y = 8 and x=2, y = 4 must satisfy the equation. So we have 16 = c(a)³ 8 = c(a)² For some constants a, c. Our goal is to find these constants. First find a and then by using a find c. Hint: Try to divide both the equations and use the law of exponents. If we found those then we have the equation y = c(a)*. b. Can there be any other exponential function that has different a and c values we obtained in (a) to satisfy the above condition (i.e passes through the two points (3, 16) and (2,8))? Why or why not? Give reasoning. c. Now let us take another set of two points, (-2,24) and (1,3). Follow the direction as in (a) to find the equation of the exponential function which passes through these two points. d. Now let us take another set of two points (2,8) and (-2, -8). Follow the direction as in (a) to find the equation of the exponential function which passes through these two points. Could we find an equation? Why or why not? Give reasoning. Hint: In an exponential function y = c(a) can a take a negative value? e. Now try to generalize what we found in this problem and come up with an answer to our original question: Given two points (1, 1) and (2, 2), can we find an exponential function y = c(a), such that it passes through both of these points? And under what conditions can the answer be "yes" to the above question? In this problem, we are going to answer the following question. Given two points (1,1) and (2,2), can we find an exponential function y = c(a), such that it passes through both of these points? If so, how many such functions can we find? a. Let us for example, take an arbitrary two points (3, 16) and (2,8). If there is an expo- nential function y = c(a)" which passes through these two points then r = 3, y = 8 and x=2, y = 4 must satisfy the equation. So we have 16 = c(a)³ 8 = c(a)² For some constants a, c. Our goal is to find these constants. First find a and then by using a find c. Hint: Try to divide both the equations and use the law of exponents. If we found those then we have the equation y = c(a)*. b. Can there be any other exponential function that has different a and c values we obtained in (a) to satisfy the above condition (i.e passes through the two points (3, 16) and (2,8))? Why or why not? Give reasoning. c. Now let us take another set of two points, (-2,24) and (1,3). Follow the direction as in (a) to find the equation of the exponential function which passes through these two points. d. Now let us take another set of two points (2,8) and (-2, -8). Follow the direction as in (a) to find the equation of the exponential function which passes through these two points. Could we find an equation? Why or why not? Give reasoning. Hint: In an exponential function y = c(a) can a take a negative value? e. Now try to generalize what we found in this problem and come up with an answer to our original question: Given two points (1, 1) and (2, 2), can we find an exponential function y = c(a), such that it passes through both of these points? And under what conditions can the answer be "yes" to the above question?