Exponential Functions and Growth Factors

This lesson explains how exponential functions grow by equal factors over equal intervals. It begins with an introduction to exponential functions and their graphical representation. The lesson then delves into understanding growth factors and provides an algebraic proof that these factors remain consistent over equal intervals. A specific function example is used to illustrate the concept further, demonstrating that exponential growth is predictable and consistent.

What is the general form of an exponential function?

• y = a + b^x

• y = a * b^x

• y = a - b^x

• y = a / b^x

In the example given, what is the growth factor from x = 2 to x = 4?

• 5

• 4

• 3

• 2

What is the y-value when x = 5 in the given example?

• 16

• 32

• 4

• 8

What does the growth factor depend on in an exponential function?

• The length of the interval

• The specific x-coordinates

• The value of b

• The value of a

How do you find the growth factor algebraically?

• By subtracting the exponents

• By adding the exponents

• By multiplying the exponents

• By dividing the exponents

What is the growth factor for an interval of length k?

• k^b

• k^a

• b^k

• a^k

In the specific function example, what is the growth factor when x increases by 5?

• 125

• 243

• 512

• 81

What is the y-coordinate when x = w in the specific function example?

• 2 * 3^w

• 3 * 2^w

• w * 3^2

• w * 2^3

What happens to y when x grows by 5 in the specific function example?

• y grows by a factor of 81

• y grows by a factor of 125

• y grows by a factor of 243

• y grows by a factor of 512

What is the purpose of the algebraic proof in the lesson?

• To show that exponential functions grow by different factors over equal intervals

• To demonstrate that exponential functions grow by equal factors over equal intervals

• To prove that exponential functions do not grow

• To illustrate that exponential functions grow by random factors