Graphs of Exponential Functions

Now that we've reviewed what exponents are and how they work, we're ready to start looking at exponential functions. Today, we'll focus on what their graphs look like. On the next page, there are 12 exponential functions ( equations and graphs are given).

(1) Divide them into two groups based on their graphs. List the groups below. How are the two groups different'? How does this show up in the equations'?

(2) Divide them into two different groups based on their graphs. List the groups below. How are the two groups different'? How does this show up in the equations'?

(3) Divide them into two groups based on their equations. List the groups below. How are the two groups different'? How does this show up in the graphs'?

( 4) Divide them into two different groups based on their equations. List the groups below. How are the two groups different'? How does this show up in the graphs'?

eSTEM Pre-Calculus Exponential Functions pg. 8

(l)x+1 2. f(x) = 2 - 1 (2) x-1

3. f(x) = 3 - 2

(3)x+2

4. 2 -2 5. J(x) = (0.4t - 4 6. J(x) = 3x-l + 2 j

, . .

' 1\ • ~ -:, \ , ~

.';. ti-- - ~1

"' ,- ,-

r ~ ~

7. J(x) = 2-x + 1 (l)x+l 8. f(x) = - 2 - 1 (2)-x-1

9. J(x) = 3 - 2

. ... 1~ - --

l

-!!J:~,:i !. ...=--- .,_l ,. ! I /

V ·-1'

I -"

,¥,

(3)x+2

10. J(x) = - 2 - 2 11. f(x) = (0.4)-x - 4 .

12. f(x) = _3x-l + 2

1,,.

: .. .__ l .._,

- - t-1 i--11 ,- -~ ; .. I l

-V

I ·+ V I r - ;. - TT T .

eSTEM Pre-Calculus Exponential Functions pg. 9

Let's see if we can come up with some ideas based on our different groupings .

How does the base in the equation of an exponential function affect the graph? ~ this -;i;,ays~

(0 l f ~~ > l, f0c-) 11..,.•.,l( r"' -tr>~-h- •

@ It o e. hN(..? l 1 -fGe) ~:AA. ~J 729v,;r)J- r /4 lj'. f.

~ ~ ~ h~~~ ~fkfro~

How does the constant in the equation of an exponential function affect the graph? Is this always true?

We saw in our function basics units that multiplying a function by a negative number on the 'inside' or 'outside' of the function caused different reflections. How can we identify reflections in the equations of exponential functions?

- f&)

eSTEM Pre-Calculus

o, ~~«1'"'<- latr),.,) ll)~.~~{.

o r ~a'" i"< X i"\ ~W\Jl -t +,

Exponential Functions pg. 10

One way we often classify exponential functions is by growth and decay.

Which of our 12 functions are examples of exponential growth'? How do you knowJ

I, 4 1 b 1

9, 9, u

Which of our 12 functions are examples of exponential decay'? How do you know'?

Some of those functions grow or decay in different ways, though, right'? We also might

want to know if growth or decay is bounded (will not pass a certain y-value) or

unbounded (will continue growing or decaying toward positive or negative infinity).

Which of the 12 functions are examples of bounded growth or decay'?

Which of the 12 functions are examples of unbounded growth or decay'?

I, 41 b I 9' {DI I( ( (7_

eSTEM Pre-Calculus Exponential Functions pg. 11

So we can classify all exponential functions into four basic shapes. List the functions that fall into the following categories, and sketch the general shape.

Unbounded Growth: 1, 4 1 0, 9, I\ Bounded Decay: '2 1 S 1 )- , 5"

Unbounded Decay: lO 1 \1.. Bounded Growth: 'a

---<E:----

eSTEM Pre-Calculus Exponential Functions pg. 12

Using what we've learned so far, sketch graphs of the following exponential functions.

f(x) = (0.25t + 2 f (x) = -(o.2sr - 2 I\"

\

' , , I ' 1 ,

- i--- ,- - ·--- -✓ ,- - / Ji

,_ -· I - ·- -1---.J-

J(x) = (0.25)-x - 2 J(x) = -(0.25)-x + 2 '

A , I

I V _,. ,

/ I/

.

l

·- - -- -l-l-·l--+--+--'1.\ --t---+-t I

- - ,-. . ---t---•---· .--

f(x) = -(4r + 3

. I \ I

' ' ...... I \_ - I I I

I°' 1\.

,__ -- ! I

I

\ ,_

' - - ·--;- ,_ ·- I

' - --· I--- I ,_ ,-. ,- -rl- ---1-- - ,-. - t I

I

Which two functions above have the same graph'? Why'?

(0:1.s )" + 7.. 0i.-,.J ( 4.)-,){ +-2

C )x \ ( -lf

(sz.1.5Y'"" f = 4f' =- \. 4)

eSTEM Pre-Calculus Exponential Functions pg. 13

Problem Set 2

Sketch graphs of the following functions and complete the statements below.

J(x) = (2y - 4 g(x) = -Gr -1

I i I ,I

I l I I I '

H- I - >- -I . I-

./ 1--- ,__

- >----I- -I- -- -- I-t--

As x ➔ oo, f ( x) ➔ ()0 As x ➔ oo, g ( x) ➔

As x ➔ -oo, J(x) ➔ -4 As x ➔ -oo, g(x) ➔

h(x) = e-x + 2 (6r· p(x) = - 7 ,I .....

\

' I'\. ! ... --

I

j

I ,__ - - --,_ ,.__

j

' _...._ ...... _ --i-- ,_ -- >- ,._ -·I-- -

As x ➔ oo, h(x) ➔ 2-As x ➔ oo, p( x) ➔

As x ➔ -oo, h(x) ➔ 00 As x ➔ -oo, p(x) ➔

eSTEM Pre-Calculus Exponential Functions pg. 14

•

Write two possible equations for the function to the right.

Write two possible equations for the function to the right.

If. 4. : cvi ~'k~ I =- I

f>evdd ~r

1·-·;--···;···--1 --7 j• ; ; ! : !

l.~ .. i_J.~L-L; : ; l :

i ................. f ........ t: .. ..... , ... :. r : :; :; •

-S 4 ..j3 -¥ ...; ~---·--·r .. ···-·r···--··!· ..... ·: : ! . i :

.;.. ...... ..;.. .J.. ..i. -: : : t :

frn .... ,f ""mf ........ 1, ... -.. i J• : i i I l

!··· ···•1· .. ·····l······+··-·: : i I i ............. --,•········;········;.-5-

! ! I t l ~ ?--•••••••!•--•t•••·" ,<OM"•! •• --:,•

! ! ! ! i ~--·- ···{ ••a.-•~········~- - ···· : ·4 ! I ·I l I : : : : I l ....... l .. ···--r--·· .... ! ........ ~··J·

:·-···+-·--+·--··+······ 11 J-..;.. ~..;,.. ~ j j ~ i i

f .. _1, ... ::P- ~-.. ::t I ! ; I I

! ~ ..... ~-~~ . ~ -·· . .-~ ....... ! ~2· : I : : I

t, .. - J .. --.J, ... , ... 1.-~.~~3 i i : i i ' ....... i ... . .... ' ........ t. .. ..... '-4

.... ----·- . ··---..... -· ' ..... , ...... -....

-f-~---f !-i ;" : ~ I t

•m•+••• +••••••+••uu,I•---••••~ : f ~ i i

l j j 4 S ·--··1··--····!··· .... "t ........ r .. u .. ,~

.... t. .. t._t .. .t .. 1 ~ ~ l I !

..... ; ........ ;--~·-•-: ..... ➔· · · · --••:

..... L ...... ~ ........ L .... .J .. ...... 1

; ' ! ! ; I I ' ! i ............... .................. i•••····-< i 1 r ; 1 : ! i ~ ~

,., ,,L ~ ... ,,,,. ~--•••••• ~ ·•••• • .. •!••••., •• ~

······+· .. ·····1········}········1········1

······! ········I ·······l ....... 1-·····I ' . . . .J -r- ~ . . i : I I : ; . , : :

.... LJ.-.. -⇒ .-LJ ; ~ ' ~ 1

··-· ! ..... , .. !·····~~; ······~:--·-··· ! l : ; I : ; ! f i ! ! ! 7 t 1 i i i i !

•••• ,! ........ • ,_,,.,.;., ...... = ••••• _ ,;

For the following table of values, sketch a graph and write a possible equation for h(x).

X -7 -4 - 1 2 5 8 11

h(x) -125 -13 1 2.75 2.969 2.996 2.999

i I

I -

eSTEM Pre-Calculus Exponential Functions pg. 15