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.

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

Lesson 3: Exponential vs. linear growth

# Intro to exponential functions

An exponential function represents the relationship between an input and output, where we use repeated multiplication on an initial value to get the output for any given input. Exponential functions can grow or decay very quickly. Exponential functions are often used to model things in the real world, such as populations, radioactive materials, and compound interest. Created by Sal Khan and CK-12 Foundation.

## Want to join the conversation?

• What does he mean by "exploding"
• He means increasing very fast.
• what happened to little x=-1?
• No one knows...
• What is the difference between exponential functions and quadratic functions
• Linear functions only have x to the first power
Quadratics have x^2 - shape is a parabola
Exponentials have x in the exponent such as 2^x, 3^x, (1/2)^x - has asymptotes where function tries to reach a y value, but never does
• how do you convert graph to equation in exponential functions?
• So the standard form for a quadratic is y=a(b)^x. So one basic parent function is y=2^x (a=1 and b=2). Learning the behavior of the parent functions help determine the how to read the graphs of related functions. You start with no shifts in x or y, so the parent funtion y=2^x has a asymptote at y=0, it goes through the points (0,1) (1,2)(2,4)(3,8),... So we find the common ratio by dividing adjacent terms 8/4=4/2=2/1=2. Thus, we find the base b by dividing the y value of any point by the y value of the point that is 1 less in the x direction which shows an exponential growth. It works the same for decay with points (-3,8). (-2,4)(-1,2)(0,1), So 1/2=2/4=4/8=1/2. It will also have a asymptote at y=0. Next, if we have to deal with a scale factor a, the y intercept will tell us that. With 2(2)^x, you double all the y values to (0,2)(1,4)(2,8)(3,16) - note that 16/8=8/4=4/2=2, so we still get the same base, but the y intercept tells us that the scale factor is 2. Similarly, if we have (0,3) and (1,6) our base is 6/3=2, but the scale factor is 3, so we have y=3(2)^x. This will work the same for decay functions, but the base will be a fraction less than 1.
So the next easiest is to shift up and down by adding a constant to the end. This can be easily be determined by a change in the asymptote. If you see an asymptote at say y=3, then "act like" this is the y axis and see how far points are away from the this line. Thus y=2^x + 3 would have points (0,4) 1 away from asymptote, (1,5) two away from asymptote, etc. Thus, you would have to do (5- 3)/(4 - 3) to get 2/1=2 as the base. This is a good introduction, which is good for all but shifts in the x direction such as y = 3 (4)^(x+2) - 5. This shifts from the origin to (-2,-5) which makes the asymptote at y=-5, but it is a little harder to determine the x axis shift back 2.
• At , why did you not solve for 3^-1, and skip right to 3^0? Is there a reason or is that just a mistake?
• Sal just made a mistake when he originally wrote the x values in the table, he skipped -1, so when he calculated y values, he skipped 3^-1.
• Whats the exponential form of 10,000
• The exponential form of 10,000 is 10^4 because it is 10 · 10 · 10 · 10. I hope this helps.
• This is a very interesting topic, i'm studying computer science and exponential function are very important for implementing algorithms.
• wha does he mean by explode
• The function with increase at a very rapid pace really quickly.