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### Course: Algebra 1>Unit 12

Lesson 7: Exponential vs. linear models

# Exponential growth & decay: FAQ

## What's the difference between exponential and linear growth?

In linear growth, we add or subtract the same amount each time period, while in exponential growth, we multiply by the same factor each time period. For example, linear growth might involve adding $5$ each day, while exponential growth might involve multiplying by $2$ each day.

## How can we tell if a graph represents exponential or linear growth?

One of the hallmarks of exponential growth is that the graph will get steeper and steeper over time. Linear growth, on the other hand, will have a constant slope.

## What's the difference between exponential growth and exponential decay?

While both involve exponential functions, exponential growth refers to when the quantity is increasing over time, while exponential decay refers to when the quantity is decreasing over time.

## How do we create an exponential function from a table or graph?

There are a few steps to this. First, identify the initial value (the $y$-intercept). Second, determine whether the function is growing or decaying. Third, find the growth or decay factor - this will be the number we're multiplying by each time period. Once we have these three pieces of information, we can write the exponential function in the form $y=a\left(b{\right)}^{x}$, where $a$ is the initial value and $b$ is the growth or decay factor.

## Where do we see exponential growth and decay in the real world?

There are many examples! Compound interest is a common example of exponential growth, while radioactive decay is a common example of exponential decay.

## Want to join the conversation?

• Is there a difference between exponential and geometric functions?
• In the case that when you say "geometric function", you are referring to a function that represents a geometric sequence, the function that represents an exponential function would look like this: f(x) = ab^x, while a geometric function would look like this: ab^x-1. Additionally, in a geometric function, the domain is any integer value that is greater than or equal to 1, while in an exponential function, the domain is all real numbers. This can make sense if you think about the fact that you cannot have a negative first or a two and a half term in a sequence of numbers.
• this is the most confusing lesson I've had on this site
someone said it evolves things you learn in algebra 2
but we're in algebra 1 for a reason
• So for me the biggest stumbling block was the common factor. What clicked - again, for me - was focusing on getting the # we multiply by with a simple equation. My first question was initial value of 4, and the next dot was 7,1. I jotted: 4x=7 (the x being more a multiplication sign in my head here). As in, 4 times what gets me to a Y value of 7? Divide 4 from each side and we x=7/4, which is what the hint/answer has for the common factor. Hope this helps anyone... as I am clawing my way through this unit slowly...
• What is the decay factor in the exponential decay function y=a(1−r)t
?
• The decay factor is the (1-r) part. Since r is a percent we subtract it from 1 to find how much is left after one round. For instance if you have 100 cookies and you eat 10% every minute, after 1 minute how many would you have?
We take the number we have and multiply by 90% to find how much is left (1 - .1 = 0.9 which is 90%)
100*(1-0.1)^1
So, unlike in growth functions where we ADD the rate to 1, in decay we SUBTRACT the rate from 1
• can a growth factor be negative
• Good question! It turns out that the answer for that is no, not really...

If the growth factor is negative, then for non-integer values of x, we would be raising a negative number to a fractional power, which does not result in a real number. For an exponential function with negative growth factor, the domain is limited to integer values of x.
This issue does not occur when the growth factor is positive, since you can plug in any positive number raised to a fractional power.

Here's an exercise: Plug in the values of (-2)^(0.5) and (2)^(0.5) into a regular calculator. Only the (-2)^(0.5) would result in a math error.

However you can multiply an exponential function by a negative number.

Happy learning.
• how do these relationships even work? like I tried to find the constant closest In the linear vs exponential growth and when I did it was wrong and the number far away from the table numbers were right
how does that make sense.
• How do I know when an exponential function is increasing or decreasing
• output values get larger as input values get larger ex: 5,25,125, 625 etc.= increasing

Output values get smaller as input values get larger ex. 100,50,25,12.5,6.25, etc. = decreasing
(1 vote)
• What type of mathematics is used in aerospace engineering? I get this could be a broad question but I'm genuinely curious.
• I got curious about functions where both the initial value and the common ratio are negative, and I stumbled across this problem.

What would the function f(x)=-5*(-2)^x look like? My calculator (Ti-84 Plus CE) is unable to graph it.

Based on my math it would go (-2, -1.25), (-1,2.5), (0,5), (1,10), (2,-20), (3,40), etc.

Is this still an exponential function? Why can't my calculator graph it? Thanks

Also, if it is an exponential function, what would you call it in regards to growth or decay? It doesn't seem to be either.
(1 vote)
• For certain problems that are not continuous for example a bank account that applies interest monthly shouldn't problems like this technically be step functions because since it only increases on certain intervals any values in between those intervals wouldn't increase, like that bank account should only increase every month so all points on the graph in-between months should be the same right? of course I think this isn't relevant to all exponentials just mainly to ones that have explicit amount of times it's compounded by.
(1 vote)
• How do we determine if f(x) is going to positive or negative infinity?
(1 vote)