If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## Get ready for Algebra 2

### Course: Get ready for Algebra 2>Unit 4

Lesson 7: Exponential growth & decay

# Exponential decay intro

Both exponential growth and decay functions involve repeated multiplication by a constant factor. However, the difference lies in the size of that factor: - In an exponential growth function, the factor is greater than 1, so the output will increase (or "grow") over time. - In an exponential decay function, the factor is between 0 and 1, so the output will decrease (or "decay") over time.

## Want to join the conversation?

• what happens if R is negative?
• I know this is old but if someone else has the same question I will answer. The equation is basically stating r^x meaning r is a base. For exponential problems the base must never be negative. I you were to actually graph it you can see it wont become exponential. just remember NO NEGATIVE BASE!
• For exponential decay, y = 3(1/2)^x but wouldn't 3(2)^-x also be the function for the y because negative exponent formula x^-2 = 1/x^2 ?
• A negative change in x for any funcdtion causes a reflection across the y axis (or a line parallel to the y-axis) which is another good way to show that this is an exponential decay function, if you reflect a growth, it becomes a decay.
• At he tells that you'll asymptote toward the x-axis. What does he mean by that? What's an asymptote?
• An asymptote is an imaginary line your function cannot cross.
• Actually first thing I thought about was y = 3 * 2 ^ - x, which is actually the same right? Using a negative exponent instead of multiplying by a fraction with an exponent.
• If the common ratio is negative would that be decay still?
• negative common ratios are not dealt with much because they alternate between positives and negatives so fast, you do not even notice it. If you have even a simple common ratio such as (-1)^x, with whole numbers, it goes back and forth between 1 and -1, but you also have fractions in between which form rational exponents. So it has not description. If the initial value is negative, it reflects the exponential function across the y axis ( or some other y = #).
• At he tells that you'll asymptote toward the x-axis. What does he mean by that? What's an asymptote?
• Asymptote is a greek word. 'A' meaning negation==NO, Symptote is derived from 'symptosis'== common case/fall/point/meet so ASYMPTOTE means no common points, which means the line does not touch the x or y axis, but it can get as near as possible.
• What is the standard equation for exponential decay?
• For exponential growth, it's generally `y = Ar^x`.
For exponential decay, it's `y = Ar^(-x)` or `y = A(1/r)^x`.

Did Sal not write out the equations in the video?
• there are some graphs where they don't connect the points. why is this graph continuous? What is the difference of a discrete and continuous exponential graph?
• I'm a little confused.

Sal says that if we have the exponential function y = Ar^x then we're dealing with exponential growth if |r| > 1

But say my function is y = 3 * (-2)^x

Well here |r| is |-2| which is 2. So I should be seeing a growth.

But if I plug in values of x I don't see a growth:

When x = 0 then y = 3 * (-2)^0 = 3
When x = 1 then y = 3 * (-2)^1 = -6
When x = 2 then y = 3 * (-2)^2 = 12
When x = 3 then y = 3 * (-2)^3 = -18

So what I'm actually seeing here is that the output is unbounded and alternates between negative and positive values.

So I suppose my question is, why did Sal say it was when |r| > 1 for growth, and not just r > 1?
• It's my understanding that the base of an exponential function is restricted to positive numbers, excluding 1.

I haven't seen all the vids yet, and can't recall if it was ever mentioned, though.

But you have found one very good reason why that restriction would be valid.