Suggested languages for you:

Americas

Europe

Problem 105

# How would you check whether data points of the form $\left(1, y_{1}\right),\left(2, y_{2}\right),\left(3, y_{3}\right)$ lie on an exponential curve?

Expert verified
To check if the data points $$(1, y_1), (2, y_2), (3, y_3)$$ lie on an exponential curve, plug the points into the exponential function $$y=ab^x$$ and solve for constants $$a$$ and $$b$$. We get the equations $$y_1=ab$$, $$y_2 = ab^2$$, and $$y_3 = ab^3$$. Solve for $$b$$ by setting the ratio of the second and first equations equal to the ratio of the third and second equations, giving us $$\frac{y_2}{y_1}=\frac{y_3}{y_2}$$. Then, solve for $$y_2^2 =y_1y_3$$, and obtain unique values for $$a = \frac{y_1^2}{y_2}$$ and $$b= \frac{y_2}{y_1}$$. Since we have unique values for $$a$$ and $$b$$, the data points lie on an exponential curve of the form $$y = ab^x$$.
See the step by step solution

## Step 1: Setup the exponential function with the data points

We have the data points $$(1, y_1), (2, y_2), (3, y_3)$$ and the exponential function $$y=ab^x$$. Let's plug the given points into the function: For point (1, $$y_1$$): $$y_1=ab^1$$ For point (2, $$y_2$$): $$y_2=ab^2$$ For point (3, $$y_3$$): $$y_3=ab^3$$

## Step 2: Solve for the constants a and b

Divide the second equation by the first equation, and the third equation by the second equation: $$\frac{y_2}{y_1} =\frac{ab^2}{ab^1} \Rightarrow \frac{y_2}{y_1}=b$$ $$\frac{y_3}{y_2}=\frac{ab^3}{ab^2} \Rightarrow \frac{y_3}{y_2}=b$$ Now since both fractions equal b, we can set them equal to each other: $$\frac{y_2}{y_1}=\frac{y_3}{y_2}$$ Solve for $$y_2$$: $$y_2^2 =y_1y_3$$

## Step 3: Verify Constants a and b

Now that we have an equation for $$y_2$$, we can plug back into the first equation and solve for $$a$$: $$y_1=ab$$ $$a= \frac{y_1}{b}=\frac{y_1}{\frac{y_2}{y_1}} = \frac{y_1^2}{y_2}$$ With $$a$$ known, we can solve for $$b$$: $$b= \frac{y_2}{y_1}$$

## Step 4: Conclusion

Since we now have unique values for constants $$a$$ and $$b$$, we can say that the data points $$(1, y_1), (2, y_2), (3, y_3)$$ lie on an exponential curve of the form $$y = ab^x$$.

We value your feedback to improve our textbook solutions.

## Access millions of textbook solutions in one place

• Access over 3 million high quality textbook solutions
• Access our popular flashcard, quiz, mock-exam and notes features

## Join over 22 million students in learning with our Vaia App

The first learning app that truly has everything you need to ace your exams in one place.

• Flashcards & Quizzes
• AI Study Assistant
• Smart Note-Taking
• Mock-Exams
• Study Planner