Intro to exponential functions (y=a^x, a^-x)
Lesson

It is not too difficult to make sense of expressions with an exponent when the exponent is an integer. For example, we interpret $2^3$23 to mean $2\times2\times2$2×2×2 and $2^4$24 to mean $2\times2\times2\times2$2×2×2×2, and so on.

But it is not so easy to give a meaning to an expression like $2^{3.5}$23.5 or, even worse, $2^{\pi}$2π.

If these expressions are to mean anything at all, we would have to agree that they are real numbers and that $2^3<2^{\pi}<2^{3.5}<2^4$23<2π<23.5<24. More generally, $2^a<2^b$2a<2b whenever $aa<b. We will want the rules for multiplying and dividing numbers with integer powers to continue to be valid when the exponents are not necessarily integers but could be any number. Recall that$a^m\times a^n=a^{m+n}$am×an=am+n$a^m\div a^n=a^{m-n}$am÷​an=amn$\left(a^m\right)^n=a^{mn}$(am)n=amn$a^{-1}=\frac{1}{a}$a1=1a$a^0=1$a0=1 for all$a\ne0$a0$a^{\frac{1}{n}}=\sqrt[n]{a}$a1n=na. In particular,$a^{\frac{1}{2}}=\sqrt{a}$a12=a. If we can convince ourselves that expressions with real number exponents do make sense, then we can construct functions which we will call exponential functions. These are of two main kinds. They have the form$y=a^x$y=ax or$y=a^{-x}$y=ax We understand$x$x to be the domain variable where$-\infty<x< and $a$a to be a constant which we call the base. Complications arise if we allow the base to be a negative number, and with zero as the base, the function would have the value zero for every non-zero $x$x. So, we will require that any base we choose for an exponential function will be a positive number.

(We will see in the examples below that we could also restrict the base to be greater than $1$1.)

We have seen that $a^0=1$a0=1 for all bases $a$a. So, the graph of every exponential function must pass through the point $(0,1)$(0,1) on the vertical axis.

Also, if $u$u and $v$v are any two numbers in the domain and $uu<v, then$a^uau<av. So, the function $a^x$ax must be an increasing function.

### Outcomes

#### 10D.QR1.04

Compare, through investigation using technology, the features of the graph of y = x^2 and the graph of y = 2^x, and determine the meaning of a negative exponent and of zero as an exponent .