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### Course: Algebra 1>Unit 12

Lesson 5: 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?

• At he tells that you'll asymptote toward the x-axis. What does he mean by that? What's an asymptote?
• An asymptote is a line (in this case a horizontal asymptote at y = 0) toward which a function approaches, but never reaches, it gets closer and closer, but can never be that number. The old riddle is that if you step toward a wall and cut each of your steps in half, will you ever reach the wall? While you could not do this in real life, in theory you could never reach the wall. that is f(x) = (1/2)^x.
• 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.
• 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?
• An asymptote is an imaginary line your function cannot cross.
• What is the standard equation for exponential decay?
(1 vote)
• 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?
• why does math exist :(
• So when you mirror exponential growth along the y-axis, you get exponential decay. What do you get when you mirror exponential growth along the x-axis?