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: Algebra I (2018 edition)>Unit 13

Lesson 3: Graphs of exponential growth

# Exponential function graph

We can graph an exponential function, like y=5ˣ, by picking a few inputs (x-values) and finding their corresponding outputs (y-values). We'll see that an exponential function has a horizontal asymptote in one direction and rapidly changes in the other direction. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• Can anyone explain to me why a negative power is always a fraction?
• Technically, an exponent expresses multiplication, and is shown with a positive number. The opposite of a positive number is (obviously) a negative number, so in keeping with the rules of exponents, this must somehow be the "opposite" of multiplication, which happens to be division. Therefore, a negative exponent is always a division (written as a fraction).
That's the reason.
• why does any number to the 0 power always become a 1? why not 0?
• As the saying goes, "Do the Math!"
At this stage you may not know all you need to with regard to the properties of exponents.
Here is one explanation that requires knowing that (x^a)/(x^b)= x^(a-b)
You know that, for example, 5/5=1, correct? It is because the numerator and denominator are equal.
Suppose you had (5^6)/(5^6). Since the numerator and denominator are equal, this is also equal to 1.
Now, using the exponential property that (x^a)/(x^b)= x^(a-b), we have
(5^6)/(5^6) = 5^(6-6) = 5^0.
And since (5^6)/(5^6) = 1 and (5^6)/(5^6) = 5^(6-6) that means 5^0 = 1 as well.

You will know lots more about exponential function when you finish this course!
• whats a slope?
• these answer above are all right.but there's another idea about slope.slope of a line can be expressed as a trigonometric function.it's the tangent of the angle formed between the x-axis and the line itself.
• We didn't get this far yet, but if the graph keeps curving as it increases, it seems like it would become somewhat circular or at least spiral.
(1 vote)
• An exponential graph doesn't keep curving. Exponential graphs just have an accelerated rate of growth/decay. There is nothing that will cause the growth/decay to bend and change direction whichis why you don't get a spiral.
• if x is infinity does that mean y is a greater exponential infinity?
• Yep!
Though we can't input an infinity into a function, the function's end behaviour (what happens as you make `x` approach `-∞` or `∞`) can be intuitively found.
As we make `x` approach `∞`, `y` also approaches `∞`, and this can be seen graphically.
• how do you do negative exponents?
(1 vote)
• The first step is to flip them for example
5^-2 would flip the -2 and because all numbers are technically fractions (5=5/ 12=12/1 this applies to any number) you would just flip it so instead of having -2 (or -2/1) you would have 1/2. so when rewritten it would be 5^1/2 then you can plug into your cauclour
• Is it ever possible for an exponential function to go into the negatives?
• The domain of an exponential function is all real numbers, but the exponential parent function has an asymptote at y=0, so it would never go into the negatives. However, all functions can be shifted down by subtracting outside the exponential function such as y = 2^x - 5.
• Why Sal drew curved lines between the points. Why not straight lines?
• What is the difference between exponents and indices?
(1 vote)
• T Ross,

Exponents are notations that indicate a base number is raised to a power or multiplied by itself a given number of times. In writing or word processing programs that allow it, exponents are written as superscript(above the base number). In a plain text editor (like this one), exponents are noted using the *^* symbol.

For example,

2^3 means 2 multiplied together 3 times: 2*2*2 = 8
x^3 means x multiplied together 4 times: x*x*x*x

Indices are a notation that indicates the position of an element in a sequence, array, or matrix. In a word processing program that allows it, indices are shown in subscript (below the name or variable assigned to the sequence). In a plain text editor, indices are indicated using the _. In the example in the video, Sal uses the sequence of numbers -2, -1, 0, 1, 2 for the x values. If you were to call this sequence X, then

X_1 = -2
X_2 = -1
X_3 = 0
X_4 = 1
X_4 = 2