Lesson 2
RCSD Geometry Local MATHEMATICS CURRICULUM
Name:___________________________________
U4
Period:________ Date:__________
GEOMETRY
Lesson 2 Exponential Functions
Learning Targets:
 I can graph and write exponential functions to model situations and describe its key features.
 I can determine when an exponential function shows growth or decay.
So far we have concentrated on __________ ______________ which are characterized by having a ________
___________ of ____________. In lesson 1, we looked at _______________ ____________ and _________.
In this lesson we will more formally introduce the concept of an ________________ function.
Example #1: Consider the exponential function 𝑓(𝑛) = 3(2)𝑛 .
Evaluate each of the following
𝑓 (2) =
𝑓 (0) =
𝑓 (−1) =
𝑓 (2) =
𝑓 (0) =
𝑓 (−1) =
Example #2: Consider the exponential
1
Function 𝑓(𝑛) = 3(2)𝑛
Evaluate each of the following
Fill in the tables for example 1 and example 2
𝑛
Use your calculator to sketch the graphs
𝑓(𝑛) = 3(2)𝑛
−2
−1
0
1
2
𝑛
1
𝑓(𝑛) = 3( )𝑛
2
−2
−1
0
1
2
Identify the 𝑦-intercept for Example 1 ________________
Example 2 ________________
Lesson 2
RCSD Geometry Local MATHEMATICS CURRICULUM
Name:___________________________________
Period:________ Date:__________
Definition: Exponential functions have the variable as the exponent.
𝒏
Remember ___ 𝒇(𝒏) = 𝒂(𝟏 + 𝒓)
𝒇(𝒏) = 𝒂(𝟏 −
Growth Factor
𝒂
𝒃
_____________
Exponential formula
Decrease/Decay factor
𝒂
𝒃 ____________
𝒇(𝒏) = _________________
Use your graphing calculator to sketch the following:
Make sure you graph and sketch one at a time.
y1  2 x
2.
1
y2   
2
3.
4.
x
y3  4 x
3
y4   
 5
𝒏
𝒓)
Example 2:
1.
U4
x
Answer these questions:
1) How can you describe the shape of the graph? _________________________________
2) Which functions show exponential growth? ____________________________________
3) Which functions show exponential decay? _____________________________________
GEOMETRY
Lesson 2
RCSD Geometry Local MATHEMATICS CURRICULUM
U4
GEOMETRY
Name:___________________________________
Period:________ Date:__________
Example 3 Complete the table for 𝑓(𝑥) = 2(3)𝑥 . Graph the function on the accompanying grid.
a. Complete the following
a: y-intercept -initial start value_______________ b: ___________ growth or decay factor
x
y
0
b.
Is this function Increasing/growth or decreasing/decay? Explain why :
c.
Fill in the table and Use your graphing calculator to sketch the graph:
1
2
3
4
Example 4 Identify the initial value in each formula below, and state whether the formula models
exponential growth or exponential decay. Justify your responses.
2 𝑡
𝑓(𝑡) = 2 (5)
5 𝑡
𝑓(𝑡) = 2 (3)
y-Int /initial :________________
y-Inter /initial:_______________
growth/decay factor:___________
growth/decay factor:___________
Exponential growth or decay?___________
Exponential growth or decay?___________
How do you know?____________________
How do you know?____________________
Example 5. The amount of Carbon 14 present after t years can be modeled by the function 𝑐(𝑡) =
10(. 98)𝑡 , where c(t) represents the amount of carbon and t represents the time, in years.
In the function c(t) , explain what 10 and .98 represent
Lesson 2
RCSD Geometry Local MATHEMATICS CURRICULUM
Name:___________________________________
U4
Period:________ Date:__________
GEOMETRY
Lesson 2 Exponential Functions
Problem Set
1.
(c) Using the points you found in (a) and (b), graph this function for the domain interval -3 < x < 3 .
Exponential Functions
f ( x)  a  b x
a: the initial amount b: growth/decay factor
If b > 1, then graph will increase
If 0 < b < 1, then graph will decrease
2. All exponential functions are in the form y = a (b)x.
a. What values of b make it an exponential growth function?_________________________
b. What values of b make it an exponential decay function?__________________________
Lesson 2
RCSD Geometry Local MATHEMATICS CURRICULUM
U4
Name:___________________________________
Period:________ Date:__________
3. Sketch an exponential growth function and an exponential decay function.
GEOMETRY
4. Identify the initial value in each formula below, and state whether the formula models exponential
growth or exponential decay. Justify your responses.
𝑓(𝑡) =
2 𝑡
(3)
3
𝑓(𝑡) = 150,000(. 65)𝑡
y-Int /initial :________________
y-Int /initial :________________
growth/decay factor:___________
growth/decay factor:___________
Exponential growth or decay?___________
Exponential growth or decay?___________
How do you know?____________________
How do you know?____________________
5. Brandon’s starting salary for his new marketing management job is $45,000. He calculates his
projected salary for the next 5 years by using the function, 𝑠(𝑡) = 45,000(1.12)𝑡 , where 𝑠(𝑡)
represents the salary amount in dollars and 𝑡 represents the time in years.
Explain what the 45,000 and the 1.12 represent.
6. The current student population of Hilton, NY is 10,000. The population can be modeled by the
function, 𝑝(𝑡) = 10,000(1.2)𝑡 .
Initial value :________________
growth/decay factor:___________
Exponential growth or decay?___________
How do you know?____________________
Lesson 2
RCSD Geometry Local MATHEMATICS CURRICULUM
Name:___________________________________
7. Given the equation 𝑓(𝑥) = 356(.75)𝑥
U4
Period:________ Date:__________
GEOMETRY
Initial value :________________ growth/decay factor:___________
Exponential growth or decay?___________
How do you know?____________________
8. Write an exponential function that shows growth. ___________________________
9. Write an exponential function that shows decay. _____________________________
10. On the set of axes below, draw the graph of over the interval
−2 ≤ 𝑥 ≤ 2,
𝑦 = 3(2)𝑥