ACTIV看TY 2亀continued
Lesson 21-3
しes§On 2書“昌
Exponential Graphs and Asymptotes
Leaming Targets:
Pacing江class period
e Determine when an exponential function is increasing or decreasing.
Chuれking theしesson
e Describe the end behavior ofexponential functions.
#1-2 #3 #4-7
⑫ Identify asymptotes of exponential functions.
Check Your Understanding
SUGGESTED LEARNING STRATEGIES: Create Representations,
Activating Prior Knowledge, CIose Reading, Vocabulary Organizer,
Think- Pair- Share, Group Presentation
Lesson Practice
・臆Bell"Ringer A`tivity
l. Graph the functionsyニ6(1.2r andy = 6(0.9)ズon a graphing
calculator or other graphing utility Sketch the results.
Ask students to find the domain and
range of each function.
2. Determine the domain and range for each function. Use interval
1・揮)二‡ [domain:allγealnum跳
notation.
D
omain
Range
exc‘Pt O; γange: all γeal ”umbeγS
(-∞,∞) (0,
∞)
a. y=6(1・ぴ
exc印t OI
2.揮)=㌔十l [domain:aliγeal
(-∞,
∞) (0,
∞)
b. yこ6(0・9)ズ
numbeγS; γangr; C,ll real肋mbeγS
A function is said to fncγeaSe ifthe y-Values increase as the x-Values increa.se.
A function is said to decrease if the y-Values decrease as the x-Values increase.
3. Describe each fu重ICtion as increasing or decreasing.
a. y二6(l・2)X
ダeater than oγequal to I]
3.揮)=X2- 2ズ十4 [domain;allγeal
numbeγS; mge‥ Oll γeai numbeγS
greateγ than oγ equal fo 3】
inc「easing
For advanced learners who wish to
further explore the Connect to AR
b. y二6(0・9)ズ
decreasing
ask students to identify the intervals
on which each function shown below
is increasing, decreasing, and/or
constant. students should also
As you learned in a previous activity, the end Z,ehaγioγ Ofa graph describes
the y-Values of the function as証ncreases without bound and as x decreases
without bound. If the end behavior approaches some constant a・ then the
identfty any functions that are strictly
monotonic.
graph ofthe function has a horizontal隼叩やtofe aty = a・
when x increases without bound,血e values ofx approach positive lnfroity’∞・
When x decreases without bound, the values of x approach negative
infinity, -∞・
4. Describe the end behavior ofeach function as x approaches ∞・ Write
・pe≧eSらしS︺盲三一<.p﹂巷言①00e二〇Uいさ乙◎
the equation for any horizontal asymptotes.
a. y= 6(1・2)ズ
[increosing lbr a/五eo/s;
As x goes to infinity, y gOeS tO infinity・
St而t/y monotonic]
b. y= 6(0・9)ズ
As x goes to infinity, y getS Ciose to O; there is a horizontaI
asymptote at y = 0"
[dec化osing佃rx < O;
1-2 C「eate Rep「esentations,
3 Åctivatiれg P「ior Knowledge.
Å`tivating P「io「 I(れOWledge,
Debriefing Students should have an
Deb「iefiれg Students should use a
intuitive knowledge of when a function
graphing calculator or other graphing
utility for Item l.
Students should note that the domain of
COnStOnt佃rO < X < 2;
increosing佃rx > 2]
is increasing and when it is decreasing.
Ask students to describe what the graph
ofan increasing function looks like and
what the graph ofa decreasing function
the負lnCtions in the problem situation
representing enlargements and
l○○ks like.
reductions are subsets of the counting
numbers, Whereas the domain of the
function in Items 2a and 2b are all real
[decreoshg佃r -2 < X < 2;
incI℃OSing lbrx < -2 andx > 2]
) numbers. Students can explore the range
of each function by using their graphing
calculators.
A⊂tivity21. Exponentia圧unctions and Graphs 329
ACT看VITY 2亀continued
4-7 ⊂『eate Represeれtations,
Think"PaiトShare. Group
Presentation, Debriefing Use
Items 4-7 to assess student understanding
Of the concepts of end behavior,
∞, -∞, and asymptotes as related to
the graphs of exponential functions.
Students should be able to answer Item 6
by referring to the graphs ofthe
負lnCtions.
Item 7 is intended to spark a dass
discussion of the features ofthe graphs of
exponential functions as related to血e
ParameterS Of the function. Some students
may not think about血e possibhity ofthe
Value of a being negative. Thy to find a
group to consider this and have them
PreSent血eir findings to血e dass.
Check Your Understanding
Debrief students’answers to these items
to ensure that they understand the
meaning of ;ncγeaSing and decγeaSing
and can identify血e domain, range, and
asymptotes of an exponential function.
Ånswe「s
8. domain (both): (-∞, ∞);
r狐ge (both): (一∞, 0)
9・輝): y aPPrOaChes -∞;
g(かy approaches O
lO・粒): y apPrOaChes O;
g(美): y apPrOaChes -∞
Students’answers to Lesson Practice
PrOblems wi11 provide you with a
formative assessment of their
understanding of the lesson concepts
and their ability to apply thelr leaming.
the problems here or use them as a
Culmination for the activity.
惟SSO悶2巨3 PRAC了IC各
ll. Increasesbecausea > Oand b > l;
y-intercept is (0, 8) because ci = 8
gives the value of the y-intercept.
12. Decreases because o > O and
O < b < 1;y-interceptis (0, 0.3)
because o = 0.3 gives the value of
the y-intercept.
13. Decreasesbecausea < O andb > l;
y-intercept is (0, -2) because cl = -2
gives the value of the y-intercept.
14. Increasesbecausec! < O andO < b < 1;
y-intercept is (0, -1) because a = -1
gives the value of the /-intercept.
15. The asymptotes are ally = 0; the
y-intercepts are all x = CZ. These
COnditions are true for all exponential
functions ofthe fomj(J*) = a(げ): When
X becomes either very la.rge or very
Small, a(のwi11 approach O; When x = 0,
.解) =a・
330 SpringBoard㊥ MathematicsAlgebra 2, ∪舶4 0 Series, Exponential and Logarithmic Functions
.pe≧劣る﹂S︺喜一﹂こく.p﹂召出eOO〇二〇U‡e乙◎
See the Activity Practice for additional
PrOblems for this lesson. You may assign