Download Lesson 3: Linear Versus Exponential Functions

Lesson 3
RCSD Geometry Local MATHEMATICS CURRICULUM
Name:___________________________________
U4
Period:________ Date:__________GEOMETRY
Lesson 3: Linear Versus Exponential Functions
Learning Targets:

I can distinguish between situations that can be modeled with linear functions and with exponential functions

I can state that linear functions grow by equal differences over equal intervals, and that exponential functions
grow by equal factors/ ratios over equal intervals.
Linear and exponential functions share many characteristics. This is because they are based on two different,
but similar, sets of principles.
LINEAR VERSUS EXPONENTIAL
Linear functions are based on repeatedly adding
the same amount
The slope (𝑚) – Constant rate of change- Common difference
Exponential functions are based on repeatedly multiplying by the same amount
The base (𝑏)- Growth or decay factor- Common Ratio
Example #1: The two tables below represent a linear function and an exponential function.
Part 1 : Which type is each function below? Explain how you arrive at your answer.
Type ______________________________________
Type _________________________________
Part 2 : Find equations in standard form for each of the functions from Example #1.
Table 1 Equation : __________________________
Part 3 : Sketch the graph of each equation
Table 2 Equation : ___________________________
Lesson 3
RCSD Geometry Local MATHEMATICS CURRICULUM
U4
Period:________ Date:__________GEOMETRY
Name:___________________________________
Example 2: Consider the linear function 𝑦 = 20𝑥 + 5 and the
exponential function 𝑦 = 5(2)2. Make a sketch of their graphs.
Which one of these grows faster?
Example 3
Example 4: Find the equation of the exponential function, in y = a(b)x form for the function given in the table
below.
Linear functions grow _________________ while exponential functions grow _________________________
Example 5: Write an equation of the function represented in the table below.
x
f(x)
–1
2
3
Type ____________________
0
1
2
3
4
2
6
18
54
162
Equation _________________________
Lesson 3
RCSD Geometry Local MATHEMATICS CURRICULUM
U4
Period:________ Date:__________GEOMETRY
Name:___________________________________
Example 6: Write an equation of the function represented in the table below.
x
–3
–2
–1
0
1
2
f(x)
5.5
5
4.5
4
3.5
3
Type ____________________ Equation _________________________
REASONING. You can determine the equation of a line or the equation of an exponential given any two points
that lie on these curves. In this exercise we will pick two special points. Consider the points (0, 5) and (1,15)
a.
Write the equation of the line that passes
between these two points in y = mx + b form.
b. Write the equation of the exponential that
passes between these two points in
c. Using your calculator, sketch the two graphs on
the axes below. Label with their equations
d. Is it fair to say that an exponential function always grows faster than a linear?
e. What can we say about an increasing exponential function when compared with an increasing linear
function?
f.
What is the difference between the way a linear function increases and the way an exponential function
increases?
Lesson Summary



Linear function of the form 𝐿(𝑥) = 𝑚𝑥 + 𝑘 grows additively by the same number over equal length intervals.
Exponential function of the form 𝑓(𝑥) = 𝑎𝑏 𝑥 grows multiplicatively by the same factor 𝑏 over equal length
intervals
An increasing exponential function will eventually exceed any linear function. Sometimes this is not apparent in
a graph displayed on a graphing calculator because the graphing window does not show enough of the graphs
for us to see the sharp rise of the exponential function in contrast with the linear function.
Lesson 3
RCSD Geometry Local MATHEMATICS CURRICULUM
U4
Period:________ Date:__________GEOMETRY
Name:___________________________________
Lesson 3: Linear Versus Exponential Functions
Problem Set
1. Look at the tables below. Which tables show a linear function and which show an exponential
function?
Table 1
Table 2
Table 3
x
y
x
y
x
y
0
2
0
20
0
81
1
8
1
17
1
27
2
32
2
14
2
9
3
128
3
11
3
3
4
512
4
8
4
1
Type __________________
Type __________________
Type __________________
Equation:_______________
Equation:_______________
Equation:_______________
2. Determine the function type: linear or exponential that can be used to model the data set and then
write an equation.
Table 1
Table 2
Table 3
x
y
x
y
x
y
0
1
0
300
0
-6
1
5
1
150
1
2
2
25
2
75
2
10
3
125
3
37.5
3
18
4
625
4
18.75
4
26
Type __________________
Type __________________
Type __________________
Equation:_______________
Equation:_______________
Equation:_______________
Lesson 3
RCSD Geometry Local MATHEMATICS CURRICULUM
U4
Period:________ Date:__________GEOMETRY
Name:___________________________________
3. Consider the points 𝐴(0,9) and 𝐵(1,3).
a. Write the equation of the line that passes between
these two points in y = mx + b form.
b. Write the equation of the exponential that passes
between these two points in
c. Using your calculator, sketch the two graphs on the
axes below. Label with their equations
4. Write an equation of the function represented in the table below.
x
–3
–2
–1
0
1
f(x)
48
24
12
6
3
Type ____________________
2
3
2
Equation _________________________
5. A lab researcher records the growth of the population of a yeast colony and finds that the population
doubles every hour.
a. Complete the researcher’s table of data:
Hours into
study
Yeast colony
population
(thousands)
0
1
2
3
4
5
b. What is the exponential function that models the growth of the colony’s population?
Lesson 3
RCSD Geometry Local MATHEMATICS CURRICULUM
Name:___________________________________
U4
Period:________ Date:__________GEOMETRY
6. Write an equation of each function represented in the table below.
x
–3
–2
–1
0
1
2
𝒈(𝒙)
–7
–3
1
5
9
13
𝑔(𝑥) = ____________________
x
–1
0
1
2
3
4
𝒉(𝒙)
1.25
5
20
80
320
1280
ℎ(𝑥) = ____________________
How does the y-intercept of the function 𝑔(𝑥) compare to the y-intercept of ℎ(𝑥) ?
How does the rate of change of the function 𝑔(𝑥) compare to the rate of change of ℎ(𝑥) ?
7. Write an exponential equation for the graph shown below. Regents 01322015
Equation:__________________________________
Explain how you determined the equation.