# 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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