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4.1 Exponential Functions

Evaluate and interpret exponential growth and decay.

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Ideas in this lesson

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Lesson brief and rules

4.1 Exponential Functions

In $f(x) = a\cdot b^{x}, b > 0$ and $b \ne 1$ . If $b > 1$ , values grow; if $0 < b < 1$ , values decay.

Remember
The exponent applies to the base before multiplication by a.

Worked example

1

01 / For

f(x)=\left(2\right)^{x}

, name the displayed base and evaluate at

-2

0

2

4

\dfrac{1}{2}

\dfrac{3}{10}

\sqrt{3}

\pi

to three decimals. Explain what distinguishes an exponential from a power function and how irrational powers are interpreted. Show the requested work; written explanations and sketches require teacher review. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.

02 /

f(x)=\left(2\right)^{x}

. The displayed constant base is

2

. Equivalently, in the form

C a^{x}

, the effective base is

a=2

. Substitute the entire input into the exponent before applying the outside multiplier.

03 /

f(-2)=\dfrac{1}{4}

(

exact

)

04 /

f(0)=1

(

exact

)

05 /

f(2)=4

(

exact

)

06 /

f(4)=16

(

exact

)

07 /

f(\dfrac{1}{2})=\sqrt{2}

, approximately

1.414

08 /

f(\dfrac{3}{10})=2^{\dfrac{3}{10}}

, approximately

1.231

09 /

f(\sqrt{3})=2^{\sqrt{3}}

, approximately

3.322

10 /

f(\pi )=2^{\pi }

, approximately

8.825

11 / An exponential has a fixed positive base other than

1

and a variable exponent. A positive base defines every real input; base

1

gives a constant, while nonpositive bases do not define this real exponential family.

12 / For an irrational exponent

\sqrt{3}

, rational inputs

1.7,1.73,1.732

give successively better approximations, not an exact finite decimal.

Worked example 2 · Define and evaluate real exponentials

For $f(x)=\left(3\right)^{- x}$ , name the displayed base and evaluate at $-2$ , $0$ , $2$ , $4$ , $\dfrac{1}{2}$ , $\dfrac{3}{10}$ , $\sqrt{3}$ , $\pi$ to three decimals. Explain what distinguishes an exponential from a power function and how irrational powers are interpreted. Show the requested work; written explanations and sketches require teacher review. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.

$f(x)=\left(3\right)^{- x}$ . The displayed constant base is $3$ . Equivalently, in the form $C a^{x}$ , the effective base is $a=0.333333333333333$ . Substitute the entire input into the exponent before applying the outside multiplier.
$f(-2)=9$ $($ exact $)$ .
$f(0)=1$ $($ exact $)$ .
$f(2)=\dfrac{1}{9}$ $($ exact $)$ .
$f(4)=\dfrac{1}{81}$ $($ exact $)$ .
$f(\dfrac{1}{2})=\dfrac{\sqrt{3}}{3}$ , approximately $0.577$ .
$f(\dfrac{3}{10})=\dfrac{3^{\dfrac{7}{10}}}{3}$ , approximately $0.719$ .
$f(\sqrt{3})=3^{- \sqrt{3}}$ , approximately $0.149$ .
$f(\pi )=3^{- \pi }$ , approximately $0.032$ .
An exponential has a fixed positive base other than $1$ and a variable exponent. A positive base defines every real input; base $1$ gives a constant, while nonpositive bases do not define this real exponential family.
For an irrational exponent $\sqrt{3}$ , rational inputs $1.7,1.73,1.732$ give successively better approximations, not an exact finite decimal.
Written-work review:

Identify the displayed positive base and distinguish the effective base from a variable-base power function.
Substitute every listed input into the whole exponent and round only final outputs to three decimals.
Explain irrational powers using rational approximations rather than asserting an exact finite input decimal.

Read the graphs

Build, compare and identify exponential graphs

First match the four panels to

y=2^{x}

y=2^{- x}

y=- 2^{x}

y=- 2^{- x}

, explaining input versus output reflection. Graph

A

. Reference plot. Curve

1

: unidentified curve.

Graph

B

. Reference plot. Curve

1

: unidentified curve.

Graph

C

. Reference plot. Curve

1

: unidentified curve.

Graph

D

. Reference plot. Curve

1

: unidentified curve.

Also compare the first curve with

y=3^{x}

on common axes and explain why

4^{\dfrac{x}{2}}

equals the first curve. Make a five-row value table and sketch

f(x)=2^{x}

and

g(x)=2 \cdot 2^{- x}

together. Describe their relationship and the reciprocal-base identity, intercepts, domain, range and asymptote. The separate marked graph below is of

y=a^{x}

and passes through

(-2,\dfrac{1}{9})

; recover its positive base. Reference plot. Curve

1

: unknown exponential.

Show the requested work; written explanations and sketches require teacher review. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.

Input	First curve	Second curve
$-2$	$\dfrac{1}{4}$	$8$
$-1$	$\dfrac{1}{2}$	$4$
$0$	$1$	$2$
$1$	$2$	$1$
$2$	$4$	$\dfrac{1}{2}$

g(x)=2f(-x)

reflects the first graph in the vertical axis and scales its heights by

2

2^{- x}

equals

\left(\dfrac{1}{2}\right)^{x}

, so the reciprocal-base curve is identical to the input-reflected curve.
Both domains are all real numbers; both ranges are positive. Their horizontal asymptote is

y=0

. The first intercept is

(0,1)

and the second is

(0,2)

a^{-2}=\dfrac{1}{9}

, with

a>0

, gives

a=3

.
Reference plot. Curve 1:

2^{x}

. Curve 2:

2 \cdot 2^{- x}

Matching the reflected-graph panels: Graph A:

y=2^{x}

; Graph B:

y=2^{- x}

; Graph C:

y=- 2^{x}

; Graph D:

y=- 2^{- x}

.
Graph B also represents

y=\left(\dfrac{1}{2}\right)^{x}

: its reciprocal-base and input-reflected formulas are coincident on those axes. The first comparison curve

y=2^{x}

also represents

y=4^{\dfrac{x}{2}}

, so these two formulas name the same plotted curve.
Reference plot. Curve 1:

2^{x}

. Curve 2:

3^{x}

For positive inputs the larger base gives the larger output; for negative inputs the ordering reverses, and both meet at

(0,1)

. Also

4^{\dfrac{x}{2}}=2^{x}

because the positive square root of the squared base is the original base.
Written-work review:

Match all four reflected graphs to their formulas, distinguishing input reflection from output reflection.
Give the five-row table, both intercepts, common real domain, positive ranges and zero asymptote.
Recover the unknown positive base from the marked point, and identify coincident reciprocal/fractional-exponent curves.
Compare distinct bases on both sides of zero and explain the reversed ordering.

Answer: $a=3$ ; relationship: input reflection and vertical scaling.

15 questions with a worked answer for each.Start practice ↗

Textbook reference: Stewart, Redlin & Watson, Precalculus: Mathematics for Calculus, 7th ed. (Cengage, 2016). The explanations and practice here are original, organized by textbook section.

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MathLab with Mr. Ahmed

4.1 Exponential Functions

Ideas in this lesson

4.1 Exponential Functions

Read the graphs

Build, compare and identify exponential graphs