In Exercises 61–64, give the equation of each exponential functio...

Question

In Exercises 61–64, give the equation of each exponential function whose graph is shown.

Graph of an exponential function with points (-4, -4/e), (0, -4), and (4, -4e).

Accepted Answer

Hey, everyone here we are asked to determine which of the following exponential functions represent the graph given below. Here in the given graph, we have three shown points, one is negative four and negative four divided by E. The second point is zero and negative four and the third point is four and negative four, E we have four answer choice options. Answer choice A Y is equal to negative four, E to the power of four X. Answer choice B Y is equal to negative two, E to the power of X divided by two. Answer choice C Y is equal to negative E to the power of X. And answer choice D Y is equal to negative four, E to the power of X divided by four. So to begin solving this problem, we need to notice that the Y coordinates of the point shown in the given graph are in terms of E. So we know that the general exponential function is given in the form Y is equal to A E to the power of B X. So now we just want to use the points on the graph to determine the variables in our formula. So we can begin by considering the 0.0 and negative four. And we can use this point to find the value of A in our formula. So here we have, from this point, we have X is equal to zero and we have Y is equal to negative four. So now just substituting in these values into our formula, we have negative four is equal to A times E to the power of B times zero. So now simplifying we have negative four is equal to A times E to the power of zero. Well, we know that E to the power of zero is just equal to one. So we actually have negative four is equal to a times one which tells us that A is just equal to negative four. Now that we have found A, we can move on to trying to find the variable B from our formula. And so we can use another point given in the graph which is four and negative for E. Now, again, here, from this point, we are given that X is equal to four and then we have Y is equal to negative four E. And then from earlier, we also found that A is equal to negative four. So now with these three values, we can just substitute them into our formula here to find our value for B. So we have negative four, E is equal to negative four times E to the power of B times four. Now just solving four B, we see both sides of the equation have negative four. So both of the negatives for negative or cancel. And so we have E is equal to E to the power of B times four or we can just have to the power of four B. And now we can simplify this expression further by taking the natural log of both sides. In doing this, we know that the natural log of E is equal to one. And so we have the one is equal to and now the natural log of E to, to the power of four B will just give us four B since the natural log makes E just one. So now solving four B, we just divide both sides by four. And we see that the value for B is 1/4. So now just using the values for A and B that we found, we can just substitute them into our formula and determine the correct function for the graph. So we have Y is equal to A, which we found to be negative four times E to the power of B X or to the power of X divided by four. So with this formula again, Y is equal to negative four E to the power of X divided by four. We are left with answer choice. D. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

In Exercises 61–64, give the equation of each exponential function whose graph is shown.

Key Concepts

Exponential Functions

Graphing Exponential Functions

Finding the Equation from Points

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