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Graphing Exponential Functions Solutions

The exponential parent function is: y=bx, b>0, b≠1.
- $b > 0 :$ The base, b, cannot be negative .
- $b \neq 1$ : The base cannot equal one because one raised to any power is one, which results in a a horizontal line .
This equation has two forms: 
- exponential growth
- exponential decay
When the exponent, x, is positive , the value of b determines whether the function represents exponential growth or decay, without creating a graph. 
And so, the value of b is also called the growth or decay factor .
Because there are an infinite number of $ℝ +$ , there are infinite parent graphs with base b .

Pay attention to $- b^{x}$ compared to ${(- b)}^{x} .$ Without parentheses, the problem would read $- 1 \cdot b^{x},$ which would mean the base is not negative.

Recall that when an exponent is negative, take the reciprocal of the base.

Growth Function	Decay Function
$b > 1 y = b^{x}, b > 0, b \neq 1$	$0 < b < 1 y = b^{x}, b > 0, b \neq 1$

Asymptote: y = 0 (x-axis) End behavior: As $x \to + \infty,$ $f (x) \to + \infty,$ and as $x \to - \infty,$ $f (x) \to 0 .$	Asymptote: y = 0 (x-axis) End behavior: As $x \to + \infty,$ $f (x) \to 0,$ and as $x \to - \infty,$ $f (x) \to + \infty .$
Domain: $\{x \| x \in ℝ\}$ Range: $\{y \| y \in ℝ, y > 0\}$

The natural exponential function, fx=ex, is a special exponential function.
- In the function, e equals 2.718281…., e is an irrational but well-rounded number.
- Because e is an irrational number, the approximation 2.718 will be used (rather than 2.718281…).

It is recommended that you use a calculator when working with e.

Note

e is defined as $e = \underset{n \to \infty}{l i m} {(1 + \frac{1}{n})}^{n} .$

Example 1

Name the value of b and if it represents growth, decay, or neither.

$f (x) = {(0.2)}^{x}$

$b = 0.2$

$q (x) = 4^{x}$

$b = 4$

$g (x) = {(0.4)}^{- x}$

$b = \frac{1}{0.4} = 2.5$

$y = {(- 8)}^{x}$

$b = - 8$

$h (x) = 10^{- x}$

$b = \frac{1}{10}$

$y = 1^{x}$

$b = 1$

$k (x) = e^{x}$

$b = e$

Note

Remember e is an irrational number.

Exponential Growth	Exponential Decay	Neither
$q (x) = 4^{x} g (x) = {(0.4)}^{- x} k (x) = e^{x}$	$f (x) = {(0.2)}^{x} h (x) = 10^{- x}$	$y = {(- 8)}^{x} y = 1^{x}$
These functions represent exponential growth because $b > 1 .$	These functions represent exponential decay because $0 < b < 1 .$	These are not exponential functions because $b = 1$ or b is a negative value.

Example 2

Graph both functions on the same coordinate plane. Name the end behavior and the domain and range.

$f (x) = 5^{x}$

$b = 5, growth$

$g (x) = {(\frac{3}{2})}^{- x}$

$b = \frac{2}{3}, decay$

Note

The negative power tells you to take the reciprocal of the base before determining if the graph will be a growth or decay function. Remember, both graphs have an asymptote at $y = 0 .$

x	f(x)	g(x)	Domain: $\{x \| x \in ℝ\}$ Range: $\{y \| y \in ℝ, y > 0\}$
–1	$\frac{1}{5}$	$\frac{3}{2}$
0	1	1
1	5	$\frac{2}{3}$

As $x \to + \infty,$ $f (x) \to + \infty,$ and as $x \to - \infty,$ $f (x) \to 0 .$

As $x \to + \infty,$ $g (x) \to 0,$ and as $x \to - \infty,$ $g (x) \to + \infty .$