# Can be 1 + 1 1

### Exponential functions of the form \$\$ y = b ^ x \$\$

What is new about exponential functions is that the variable (the \$\$ x \$\$) is in the exponent, i.e. the exponent.

Example: \$\$ y = 2 ^ x \$\$.

For example, if \$\$ x \$\$ is \$\$ 4 \$\$, the function value is \$\$ = 2 ^ 4 = 2 * 2 * 2 * 2 = 16 \$\$.

These functions, like others, can be represented using graphs or tables of values.

### The table of values ​​of \$\$ y = 2 ^ x \$\$

You will now see a table of values ​​for this function: The function value \$\$ y \$\$ doubles with every step to the right in the table.

The following power laws are helpful for calculating the \$\$ y \$\$ values:

For powers \$\$ a ^ b \$\$ with \$\$ a \ in \ mathbb {R} \$\$ and \$\$ b \ in \ mathbb {Z} \$\$ applies in particular:

\$\$ a ^ -b = 1 / {a ^ b} \$\$ and \$\$ a ^ 0 = 1 \$\$.

### The graph of \$\$ y = 2 ^ x \$\$

If you enter the \$\$ x \$\$ and \$\$ y \$\$ values ​​in the coordinate system, you will get the following graph: As you can see the graph is increasing. The function describes exponential things growth. If the graph fell, you'd have exponential Decay to do.

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### Definition: Exponential functions of the form \$\$ y = b ^ x \$\$

A function with the equation \$\$ y = b ^ x \$\$ with \$\$ b> 0 \$\$ and \$\$ b \ ne1 \$\$ is called an exponential function based on \$\$ b \$\$.

The \$\$ b \$\$ is also called the growth or decay factor.

The case \$\$ b = 1 \$\$ is also excluded here because \$\$ b = 1 \$\$ there stands \$\$ y = 1 ^ x \$\$. The result of this is always \$\$ 1 \$\$, since here only the number \$\$ 1 \$\$ is multiplied by itself as often as required.

And why should \$\$ b \$\$ not be negative?

This is the table of values ​​for \$\$ y = (- 2) ^ x \$\$:

\$\$ x \$\$ \$\$-3\$\$\$\$-2\$\$\$\$-1\$\$\$\$ 0\$\$\$\$1\$\$\$\$2\$\$\$\$3\$\$
\$\$ y \$\$ \$\$-1/8\$\$\$\$1/4\$\$\$\$-1/2\$\$\$\$1\$\$\$\$-2\$\$\$\$4\$\$\$\$-8\$\$

You can see that the \$\$ y \$\$ values ​​jump from \$\$ + \$\$ to \$\$ - \$\$. Imagine that in the coordinate system:
You cannot connect the points with a line.

Such functions do not occur in everyday life, so you do not have to investigate them. :-)

### Properties of exponential functions of the form \$\$ y = b ^ x \$\$

Look at the graphs and discover general properties. You can see from the graph:

• The graph falls for \$\$ b \$\$ between \$\$ 0 \$\$ and \$\$ 1 \$\$ (exponential decay) and rises for \$\$ b \$\$ greater than \$\$ 1 \$\$ (exponential growth).

• The graphs are all above the \$\$ x \$\$ axis. So there are no negative \$\$ y \$\$ values. The 0 is also not a \$\$ y \$\$ value.

• The graphs cling to the \$\$ x \$\$ axis.

• All graphs run through the point \$\$ P (0 | 1) \$\$.

• If the base \$\$ b \$\$ is very close to \$\$ 1 \$\$, the graph resembles a straight line with \$\$ y = 1 \$\$.

### Determine the functional equation at the given point

You have already seen that the graphs of all exponential functions of the form \$\$ y = b ^ x \$\$ intersect at the point \$\$ (0 | 1) \$\$. In fact, they only have this one in common! The following calculation makes this plausible.

You have given the point \$\$ (7 | 3) \$\$ and are supposed to give the corresponding functional equation:

\$\$ y = b ^ x \$\$ | Insert point

\$\$ 3 = b ^ 7 \$\$ | \$\$ root7 \$\$

\$\$ b = \$\$ \$\$ root7 3≈ 1.17 \$\$

\$\$ ⇒ y ≈ 1.17 ^ x \$\$

The last function described contains this point, and only this one! As soon as \$\$ x \$\$ or \$\$ y \$\$ changes, the \$\$ b \$\$ also changes.

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