Deciding whether its a exponential growth or decay

Question

Given the following exponential function, identify whether the change represents growth or decay, and determine the percentage rate of increase or decrease.

y=64(0.13)^x

Answer 1

y=64(0.13)^x

Let us take any value for x, and for simplicity, Take value of x equal zero,

at x=0

y = 64(0.13)^0

y= 64 * 1 =64 ,

when x start to increase, you can take the limt of y as x goes to ∞ .

lim_{x → ∞} y = 64 (0.13)^^∞ = 0 , hint : 0.13 = 13/ 100, when we raise to high power the

denominator start to rise faster than the numerator

and that's why the result goes to zero.

So, the graph will decay as x increase.

To calculate the rate of decrease you can take the derivative of y:

y = 64 (0.13)^x

y = 64 b^{x ,} where b = 0.13, d( b^x )/dx = ln b* b^x

dy /dx = 64( ln b) b^x = 64 ( -2.0422) (0.13)^x = - 130.574 * (0.13)^x

the minus sign indicate that we have decreasing function.

Answer 2

Hi Hannah!

This is a two-part question: Is it exponential growth or decay? From there, we must ask: What is the percentage increase or decrease?

To start, we must examine which parts of the function are important. In this case, because we are looking at exponential growth and decay, this deals with the part of the function that is inside the parentheses.

Original f(x): y=64(0.13)^x

What we want to focus on: (0.13)

In an exponential equation like this, if the number inside the parentheses is between 0 and 1, it is a decaying function and gets significantly smaller over time. If the number in the parentheses is greater than 1, then it will be exponentially growing. In this case, 0.13 is less than one so the function is DECAYING

For the percentage decrease (decreasing because we have shown that it is a decaying function), you use the formula y=(1-b)^x

While this may look confusing, it's actually quite simple! The "b" term is just whatever was in the parentheses of the original equation! For us, that would be 0.13

So, plugging it in you would get something that looks like this:

y=(1-.13)

To solve, subtract the numbers and you get 0.87.

In percentage form, that would be 87% and you have found the percent decrease of this function!

Hope this helped!

Answer 3

| r | > 1, growth

| r | < 1, decay