TBIL-Precal Graphs of Exponential Functions (EL2)

Its domain is $(- \infty, \infty)$ and its range is $(0, \infty) .$
It is an exponential growth function; that is it is increasing on $(- \infty, \infty) .$
There is a horizontal asymptote at $y = 0 .$ There is no vertical asymptote.
There is a $y$ -intercept at $(0, 1) .$ There is no $x$ -intercept.

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Activity 5.2.3.

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Consider the function

g (x) = {(\frac{1}{2})}^{x} .

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(a)

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Fill in the table of values for

g (x) .

Then plot the points on a graph.

$x$	$g (x)$
$- 2$
$- 1$
$0$
$1$
$2$

Answer.

$x$	$g (x)$
$- 2$	$4$
$- 1$	$2$
$0$	$1$
$1$	$\frac{1}{2}$
$2$	$\frac{1}{4}$

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(b)

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What seems to be happening with the graph as

x

goes toward infinity? Plug in large positive values of

x

to test your guess, then describe the end behavior.

As $x \to \infty,$ $g (x) \to - \infty .$
As $x \to \infty,$ $g (x) \to - 2 .$
As $x \to \infty,$ $g (x) \to 0 .$
As $x \to \infty,$ $g (x) \to 2 .$
As $x \to \infty,$ $g (x) \to \infty .$

Answer.

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(c)

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What seems to be happening with the graph as

x

goes toward negative infinity? Plug in large negative values of

x

to test your guess, then describe the end behavior.

As $x \to - \infty,$ $g (x) \to - \infty .$
As $x \to - \infty,$ $g (x) \to - 2 .$
As $x \to - \infty,$ $g (x) \to 0 .$
As $x \to - \infty,$ $g (x) \to 2 .$
As $x \to - \infty,$ $g (x) \to \infty .$

Answer.

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(d)

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Complete the graph you started in Task 5.2.3.a, connecting the points and including the end behavior you’ve just determined.

Answer.

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(e)

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What are the equations of the asymptote(s) of the graph?

Answer.

y = 0

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(f)

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Find the domain and range of

f (x) .

Write your answers using interval notation.

Answer.

Domain:

(- \infty, \infty),

Range:

(0, \infty)

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(g)

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Find the interval(s) where

f (x)

is increasing and the interval(s) where

f (x)

is decreasing. Write your answers using interval notation.

Answer.

Increasing: nowhere, Decreasing:

(- \infty, \infty)

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Activity 5.2.4.

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Consider the two exponential functions we’ve just graphed:

f (x) = 2^{x}

and

g (x) = {(\frac{1}{2})}^{x} .

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(a)

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How are the graphs of

f (x)

and

g (x)

similar?

Answer.

Answers could include basic shape, asymptote, domain, range.

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(b)

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How are the graphs of

f (x)

and

g (x)

different?

Answer.

Answers could include reflection over

y

-axis, one is increasing, one is decreasing.

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Remark 5.2.5.

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We can now update Remark 5.2.2 so that it includes all values of the base of an exponential function.

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The graph of an exponential function

f (x) = b^{x}

has the following characteristics:

Its domain is $(- \infty, \infty)$ and its range is $(0, \infty) .$
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Remember, exponential functions are only defined when $b > 0$ and $b \neq 1$ as we saw in Definition 5.1.4.
If $b > 1,$ $f (x)$ is increasing on $(- \infty, \infty)$ and is an exponential growth function. If $0 < b < 1,$ $f (x)$ is decreasing on $(- \infty, \infty)$ and is an exponential decay function.
There is a horizontal asymptote at $y = 0 .$ There is no vertical asymptote.
There is a $y$ -intercept at $(0, 1) .$ There is no $x$ -intercept.

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Activity 5.2.6.

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Let’s look at a third exponential function,

h (x) = 2^{- x} .

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(a)

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Before plotting any points or graphing, what do you think the graph might look like? What sort of characteristics might it have?

Answer.

With the

2

as the base, students may assume exponential growth. But the negative in the exponent may change their mind if they remember transformations!

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(b)

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Fill in the table of values for

h (x) .

Then plot the points on a graph.

$x$	$h (x)$
$- 2$
$- 1$
$0$
$1$
$2$

Answer.

$x$	$h (x)$
$- 2$	$4$
$- 1$	$2$
$0$	$1$
$1$	$\frac{1}{2}$
$2$	$\frac{1}{4}$

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(c)

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This function

h (x)

looks to be the same as a function we looked at previously. Use properties of exponents to rewrite

h (x)

in a different way.

Answer.

2^{- x} = (2^{- 1})^{x} = {(\frac{1}{2})}^{x}

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(d)

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In addition to plotting points, we can use transformations to graph. If we consider

f (x) = 2^{x}

to be the parent function, what transformation is needed to graph

h (x) = 2^{- x} ?

A vertical stretch.
A horizontal stretch.
A reflection over the $x$ -axis.
A reflection over the $y$ -axis.

Answer.

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Remark 5.2.7.

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For a reminder of transformations, see Section 2.4 and the following definitions:

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Activity 5.2.8.

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Let

f (x) = 4^{x} .

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(a)

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Graph

f (x) .

Answer.

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(b)

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Match the following functions to their graphs.

$g (x) = - 4^{x}$
$h (x) = 4^{- x}$
$j (x) = 4^{x + 1}$
$k (x) = 4^{x} + 1$

Answer.

$g (x) = - 4^{x}$ is D
$h (x) = 4^{- x}$ is B
$j (x) = 4^{x + 1}$ is C
$k (x) = 4^{x} + 1$ is A

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(c)

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Find the domain, range, and equation of the asymptote for the parent function

(f (x))

and each of the four transformations

(g (x), h (x), j (x), and k (x)) .

Answer.

f (x) :

Domain: $(- \infty, \infty)$
Range: $(0, \infty)$
Asymptote: $y = 0$

g (x) :

Domain: $(- \infty, \infty)$
Range: $(- \infty, 0)$
Asymptote: $y = 0$

h (x) :

Domain: $(- \infty, \infty)$
Range: $(0, \infty)$
Asymptote: $y = 0$

j (x) :

Domain: $(- \infty, \infty)$
Range: $(0, \infty)$
Asymptote: $y = 0$

k (x) :

Domain: $(- \infty, \infty)$
Range: $(1, \infty)$
Asymptote: $y = 1$

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(d)

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Which of the transformations affected the domain of the exponential function? Select all that apply.

A vertical shift.
A horizontal shift.
A reflection over the $x$ -axis.
A reflection over the $y$ -axis.
None of these.

Answer.

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(e)

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Which of the transformations affected the range of the exponential function? Select all that apply.

A vertical shift.
A horizontal shift.
A reflection over the $x$ -axis.
A reflection over the $y$ -axis.
None of these.

Answer.

A and C

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(f)

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Which of the transformations affected the asymptote of the exponential function? Select all that apply.

A vertical shift.
A horizontal shift.
A reflection over the $x$ -axis.
A reflection over the $y$ -axis.
None of these.

Answer.

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Activity 5.2.9.

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Consider the function

f (x) = e^{x} .

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(a)

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Graph

f (x) = e^{x} .

First find

f (0)

and

f (1) .

Then use what you know about the characteristics of exponential graphs to sketch the rest. Then state the domain, range, and equation of the asymptote. (Recall that

e \approx 2.72

to help estimate where to put your points.)

Answer.

Domain: $(- \infty, \infty)$
Range: $(0, \infty)$
Asymptote: $y = 0$

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(b)

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Sketch the graph of

g (x) = e^{x - 2}

using transformations. State the transformation(s) used, the domain, the range, and the equation of the asymptote.

Answer.

Transformation: shift right 2
Domain: $(- \infty, \infty)$
Range: $(0, \infty)$
Asymptote: $y = 0$

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(c)

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Sketch the graph of

h (x) = - 3 e^{x}

using transformations. State the transformation(s) used, the domain, the range, and the equation of the asymptote.

Answer.

Transformations: vertical stretch of 3, reflection over $x$ -axis
Domain: $(- \infty, \infty)$
Range: $(- \infty, 0)$
Asymptote: $y = 0$

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(d)

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Sketch the graph of

g (x) = e^{- x} - 4

using transformations. State the transformation(s) used, the domain, the range, and the equation of the asymptote.

Answer.

Transformations: reflection over $y$ -axis, shift down 4
Domain: $(- \infty, \infty)$
Range: $(- 4, \infty)$
Asymptote: $y = - 4$

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Activity 5.2.10.

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Graph each of the following exponential functions. Include any asymptotes with a dotted line. State the domain, the range, the equation of the asymptote, and whether it is growth or decay.

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