STAAR Exponential Growth and Decay Word Problems

Exponential Functions

​Definition

An exponential function is a function where the independent variable for x is written as an exponent.

Multiple Choice

Which of the following is an example of an exponential function?

• $$f\left(x\right)=\ 3x\ +\ 2$$

• $$f\left(x\right)=\ 2\left(3\right)^x$$

• $$f\left(x\right)=3x^2+2x\ +\ 2$$

Final value → f(x)

Initial value → a

Rate → b

timE → x

Equation: f(x)=a(b)^x

Think of the word fire to remember the different parts of the equation for these new types of functions.

Memory Trick

Exponential Functions

Match

Match the following

final value

initial value

rate

time

• final value

• initial value

• rate

• time

Exponential Functions

​Function Parts

The next few slides will talk about different parts of the equation:

• Final Value

• Initial Value

• Rate

• Time

Multiple Choice

What is the initial value in

$$32\ =\ 4\left(2\right)^3$$

• 32

• 4

• 2

• 3

Multiple Choice

What is the rate in

$$32\ =\ 4\left(2\right)^3$$

• 32

• 4

• 2

• 3

f(x) = a(b)^x

Initial Value

The value for a can be found by the following patterns / keywords:

• Tables → the y-value list first or next to where x = 0

• Ordered pairs → (0, a)

• Graphs → the y-intercept

• Word problems → initial, begin, start, currently

For Quiz 1 no need to focus on ordered pairs and graphs

Multiple Choice

What is the initial value from the table?

(hint: look for the y-intercept value)

• 2023

• 2000

• 2100

• 2205

• 2315.25

Multiple Choice

What is the initial value from the table?

(hint: look for the y-intercept value)

• 80

• 40

• 20

• 10

• 0

Multiple Choice

Forest Park started 2022 with 50,000 trees. Total trees will decrease 5% each year.

(What is the initial value?)

• 2022

• 50,000

• 5%

• 0.05

Multiple Choice

A Petri dish currently has 48 bacteria. Each hour, the total number doubles. The bacteria are allowed to grow for one full day.

(What is the initial value?)

• 48 bacteria

• 2

• 1 day

• 24 hours

f(x) = a(b)^x

Rate

The value for b can be found by the following patterns / keywords:

• Word problems with no percentage in question

  • Keywords → double, triple, quadruple, etc.

• Word problems with percentage in question

  • Keywords → growth, increase, decay, decrease, eliminate

For Quiz 1 focus on getting value from word problems

Double = 2

Triple = 3

Quadruple = 4

Rate Keywords

The number in parenthesis will be directly given with keywords in the word problem

f(x) = a(b)^x

Finding rate from word problems that do NOT mention a percentage

Some text here about the topic of discussion

Multiple Choice

A Petri dish currently has 48 bacteria. Each hour, the total number doubles. The bacteria are allowed to grow for one full day.

(What is the rate?)

• 48 bacteria

• 2

• 1 day

• 24 hours

Convert % into decimal

• Divide % by 100 → r

Rewrite b as growth or decay

• b = (1 + r)

• b = (1 - r)

Steps to Follow

Keywords increase & growth tell you to write b as (1 + r)

---------------------------------------

Keywords decrease & decay tell you to write b as (1 - r)

f(x) = a(b)^x

Finding rate from word problems that DO mention a percentage

Some text here about the topic of discussion

Match

Match rate when written as a % and decimal

• 3%

• 30%

• 5%

• 50%

0.03

0.30

0.05

0.50

• 0.03

• 0.30

• 0.05

• 0.50

Multiple Choice

Forest Park started 2022 with 50,000 trees. Total trees will decrease 5% each year.

(What is the rate?)

• 5%

• 0.05

• (1 - 0.5)

• (1 - 0.05)

• (1 + 0.05)

f(x) = a(b)^x

Time

The of the value for x as the difference of two values

• x = end time - start time

• The end time will always be stated in the word problem

• The start time may be stated in the word problem OR in a table

  • Important: assume start time is zero if not mentioned

For Quiz 1 focus on getting value from word problems

Multiple Choice

A Petri dish currently has 48 bacteria. Each hour, the total number doubles. The bacteria are allowed to grow for one full day.

(What is the time?)

• 0

• 1 day

• 24 hours

• 2

Multiple Choice

Jacob has a savings account that earns 5% interest annually. How much will be in the savings account by 2030?

• 2023

• 2030

• 7

• 2000

• 10

1) Annotate keywords

2) Write known variables

• Final value

• Initial value

• Rate

• Time

3) Use equation: f(x) = a(b)^x

Steps to Follow

Word Problems:

Creating Functions

1) Enter table into Desmos (@ least 3 rows of data)

2) Enter function into Desmos with regression symbols

• ~, y₁, and x₁

3) Interpret R² value (anywhere from 0 to 1)

• Function that is good fit for table has value near 1

• Function that is bad fit for table has a value near 0

Steps to Follow

Tables:

Creating Functions with Desmos

Build a Function for table below:

Tables:

Creating Functions with Desmos (example)

​Step 1 Step 2

​y = 2000(1.05)^x