How to Determine Values in Exponential Functions

TL;DR

To find the values of a and r in an exponential function, start by calculating the common ratio r from the ratios of given values of h(n). Then, use either a tabular method or solve an equation with known values to determine the initial value a. Both methods will yield the same results.

Transcript

• [Voiceover] Let's say that we have an exponential function, h of n, and since it's an exponential function it's going to be in the form a times r to the n, where a is our initial value and r is our common ratio, and we're going to assume that r is greater than zero. And they've given us some information on h of n. We know that when n is equal to ... Read More

Key Insights

• 🥳 Exponential functions can be written in the form a times r to the n, where a is the initial value and r is the common ratio.
• 🥳 The common ratio r can be found by comparing the ratios of h(n) for different values of n.
• ❓ The initial value a can be found by using a tabular method or solving an equation.
• 🥳 The values of a and r can be solved for if the values of h(n) and the common ratio are known.
• 🫚 The principal root of a given fraction can be determined to find the value of r.
• 🚃 There are multiple ways to find the values of a and r, but they yield equivalent results.
• 🥳 Understanding the common ratio and initial value in an exponential function is crucial for solving problems involving exponential growth or decay.

Questions & Answers

Q: How can we find the common ratio r in an exponential function?

By comparing the ratios of h(n) for different values of n, such as h(4)/h(2), the common ratio r can be determined.

Q: How can we find the initial value a in an exponential function?

The initial value a can be found by either using a tabular method or solving an equation. In the tabular method, a table is set up comparing h(n) for different values of n, and the values are calculated using the common ratio. Solving an equation involves setting up an equation with known values of h(n) and r, and solving for a.

Q: Can we solve for a if we know the function for a given n and r?

Yes, if we know the function for a given n and r, we can solve for a. By rearranging the exponential function equation and substituting the known values, we can find the value of a.

Q: What are the conditions for finding the values of a and r?

The given exponential function must be in the form a times r to the n, and the common ratio r must be greater than zero.

Summary & Key Takeaways

• The content explains how to find the values of a and r in the exponential function a times r to the n, based on given values of h(n).

• By comparing the ratios of h(n) for different values of n, the common ratio r can be determined.

• The initial value a can be found by either using a tabular method or solving an equation.