Just from looking at \(f(t) = a \cdot b^t\), how do you know whether the quantity is growing or shrinking — without calculating a single value?
Learning Targets
I can explain (in writing) the meaning of different constants in an expression representing an exponential function. I can interpret equations and graphs that represent exponential functions in context.
This is an optional review lesson. Students revisit exponential growth and decay across multiple representations: descriptions, tables, graphs, and equations. They also evaluate exponential functions at negative inputs and interpret the results in context (MP2, MP7). Standards: HSA-SSE.A.1, HSF-LE.B.5, HSF-IF.B.4, HSF-LE.A.2
Warm-Up ⏱ 5 min
One Fourth at a Time
This warm-up prompts students to interpret the meaning of each part of an expression that represents a recursively defined geometric sequence. Activate prior knowledge about geometric sequences and repeated multiplication by the same factor. Cold-call 2–3 students to share their table before the partner discussion.
Priya borrowed $160 from her grandmother with a promise to pay it back a little at a time. Each month, she pays back \(\frac{1}{4}\) of what she still owes.
Work on your own, then share with a partner
Complete the table showing how much Priya pays and owes each month (months 0 through 3).
Discuss with a partner why the expression \(160 \cdot \left(\frac{3}{4}\right)^3\) represents the balance Priya owes her grandmother at the end of the third month.
Try Saying
“The expression \(160 \cdot \left(\frac{3}{4}\right)^3\) makes sense because ___.”
Warm-Up Synthesis ⏱ 3 min
What does each part of the expression mean?
Display a completed table. Make sure students can articulate the meaning of each piece: 160 is the initial amount; \(\frac{3}{4}\) is the fraction of the previous balance still owed; the exponent 3 is the number of months. Emphasize the term "exponentially" — the amount changes by the same factor each time.
The expression \(160 \cdot \left(\frac{3}{4}\right)^3\) represents the amount Priya owes after 3 months. Each part of this expression has a specific meaning.
“What does the 160 represent?”160 represents the initial amount, in dollars, that Priya owed her grandmother.
“What does the \(\frac{3}{4}\) represent?”\(\frac{3}{4}\) is the fraction of the previous balance that is still owed after Priya makes each monthly payment. She pays \(\frac{1}{4}\), so \(\frac{3}{4}\) remains.
“What does the exponent of 3 mean?”The exponent 3 is the number of months Priya has made payments.
We can describe the amount owed as changing exponentially because it is changing by the same factor each time.
click to advance discussion ▶
Sample Responseclick to reveal ▶
Month
Amount Paid ($)
Amount Owed ($)
0
0
160
1
40
120
2
30
90
3
22.50
67.50
Try Saying
“I think the amount changes exponentially because ___.”
“The factor \(\frac{3}{4}\) appears because ___.”
MLR6 · Three Reads ⏱ 5 min
Climbing Cost
The tuition at a college has been increasing by the same percentage since the year 2000. The tuition was $30,000 in 2012, $31,200 in 2013, and $32,448 in 2014.
Read 1
What is this situation about?
Try Saying
“This is about ___ and ___.”
Read 2
What quantities can be counted or measured?
Try Saying
“I see the quantities ___ and ___.”
Read 3
The equation \(f(t) = 30{,}000 \cdot (1.04)^t\) represents the cost of tuition, in dollars, as a function of \(t\), the number of years since 2012. Explain what the 30,000 and 1.04 tell us about this situation.
What strategy could you try first?
Try Saying
“I think ___ works because ___.”
MLR6: Three Reads. Read the context aloud. Click to reveal each read's focus one at a time. After Read 1: cold-call 2–3 students — "What is this about?" (the increasing cost of college tuition). After Read 2: pair-share quantities (cost of tuition, number of years since 2012 or 2000). After Read 3 (question revealed): give 1 min individual think time before partner work begins. If students do not recall what "increasing by the same percentage" means, clarify that a percent increase means the college applies the same percent increase each year. If students need help recalling how percent change works, pause the class after 2–3 min and have students share the meaning behind the 1 and 4 in 1.04.
Activity 1 · Work Time ⏱ 10 min
Climbing Cost
🧑🏫 Look for: students who connect 1.04 to a 4% increase; students who struggle with negative exponents in problem 4. Common error on problem 3: students may multiply 30,000 by 3 instead of raising 1.04 to the power of 3. Common error on problem 4: students may write \(30{,}000 \cdot (1.04)^5\) instead of \(30{,}000 \cdot (1.04)^{-5}\).
If Students Are Stuck
This is the first example of many in which an expression only approximately models a value — tuition, in this case. In reality, the tuition would probably be rounded. If students are confused by the complicated number that results when they use a calculator to find the tuition from 5 years ago, consider asking: "Can you explain how you determined the tuition value for 2007?" "What is the same and what is different about the tuition value you found and the values given from 2012, 2013, and 2014?"
Amplify · Activity Builder · Type-in
Setup (do once):
1. Add a Text component with all student-facing text.
2. Add a Math Response component → name it answer
3. Add a Note component → click </> → paste CL
5. Straight quotes only
Problem 1: Explain 30,000 and 1.04
content:
when answer.submitted
"Responses will vary. Look for: 30,000 is the tuition in 2012; 1.04 is the growth factor (4% increase each year)."
otherwise ""
Problem 2: Percent increase
content:
when answer.submitted and answer.latex = "4"
"Correct! The percent increase is 4% per year."
when answer.submitted and answer.latex = "4%"
"Correct! The percent increase is 4% per year."
when answer.submitted and answer.latex = "0.04"
"Almost! 0.04 is the decimal form. What percent does that represent?"
when answer.submitted and answer.latex = "1.04"
"Not quite — 1.04 is the growth factor. The percent increase is just the part above 100%. What is 1.04 minus 1?"
when answer.submitted
"The growth factor is 1.04. Subtract 1 to find the percent increase as a decimal, then convert to a percent."
otherwise ""
Problem 3: f(3)
content:
when answer.submitted and answer.latex = "33745.92"
"Correct! f(3) = 30000(1.04)^3 = $33,745.92"
when answer.submitted and answer.latex = "33,745.92"
"Correct! f(3) = 30000(1.04)^3 = $33,745.92"
when answer.submitted and answer.latex = "33745.9"
"Correct! f(3) = 30000(1.04)^3 is approximately $33,745.92"
when answer.submitted
"Find f(3) = 30000 times (1.04)^3. Use a calculator to evaluate (1.04)^3 first."
otherwise ""
Problem 4: Expression for 2007
content:
when answer.submitted and answer.latex = "30000(1.04)^{-5}"
"Correct! 2007 is 5 years before 2012, so t = -5."
when answer.submitted and answer.latex = "30000 \\cdot (1.04)^{-5}"
"Correct! 2007 is 5 years before 2012, so t = -5."
when answer.submitted and answer.latex = "30000(1.04)^5"
"Good start — but 2007 is BEFORE 2012. What sign should the exponent have?"
when answer.submitted and answer.latex = "30000 \\cdot (1.04)^5"
"Good start — but 2007 is BEFORE 2012. What sign should the exponent have?"
when answer.submitted
"2007 is how many years before 2012? Use that as the value of t in f(t) = 30000(1.04)^t."
otherwise ""
Problem 5: Tuition in 2007
content:
when answer.submitted and answer.latex = "24657.81"
"Correct! Tuition in 2007 was approximately $24,657.81."
when answer.submitted and answer.latex = "24,657.81"
"Correct! Tuition in 2007 was approximately $24,657.81."
when answer.submitted and answer.latex = "24657.8"
"Correct! Tuition in 2007 was approximately $24,657.81."
when answer.submitted
"Evaluate 30000 times (1.04)^(-5) using a calculator. What do you get?"
otherwise ""
The tuition at a college has been increasing by the same percentage since 2000. The tuition was $30,000 in 2012, $31,200 in 2013, and $32,448 in 2014. The equation \(f(t) = 30{,}000 \cdot (1.04)^t\) represents the cost of tuition, in dollars, as a function of \(t\), the number of years since 2012.
Work with your partner
Explain what the 30,000 and 1.04 tell us about this situation.
What is the percent increase in tuition from year to year?
What does \(f(3)\) mean in this situation? Find its value and show your reasoning.
Write an expression to represent the cost of tuition in 2007.
How much did tuition cost that year?
★ Challenge
(Challenge 1) In what year did tuition first exceed $25,000? Write an inequality and solve it, or use a table of values.
(Challenge 2) Another college had tuition of $28,000 in 2012 and increases by 5.5% per year. In what year does this college's tuition overtake the first college's tuition?
Activity 1 · Synthesis ⏱ 5 min
Growth Factor and Negative Exponents
Highlight: (1) how to read percent increase from the growth factor; (2) the meaning of negative exponents as "going back in time." Select a student who connected 1.04 to 104% = 100% + 4% to share first, then a student who used negative exponents for 2007.
The purpose of this discussion is to make sure everyone can interpret the growth factor and use negative exponents to find past values.
“How can we tell what the percent of increase is?”The 1.04 from the function can be interpreted as 104%, which is 100% of the tuition plus an increase of 4%.
“If the equation were not given, can we still tell that the tuition is growing by 4%? How?”Yes. We can find the quotient of the tuition costs in consecutive years: \(\frac{31{,}200}{30{,}000} = 1.04\) and \(\frac{32{,}448}{31{,}200} = 1.04\). Each is an increase of 4%.
“Why does \(-5\) make sense as the input for 2007?”Since the input \(t\) is years since 2012, finding the tuition in 2007, which is 5 years before 2012, means using an input value of \(-5\).
“Are the tuition costs changing exponentially? How do we know?”Yes. Each year it is increasing by the same percentage (or by the same factor).
click to advance discussion ▶
Try Saying
“I know the growth rate is ___% because ___.”
“A negative exponent means ___ because ___.”
Math As A Second Language
Exponential Growth Function
Math · We Say · Meaning
★ Math (given)
\(f(t) = 30{,}000 \cdot (1.04)^t\)
▲ We Say
“f of t equals 30,000 times 1.04 to the power of t”
● Meaning
An exponential function where 30,000 is the initial value and 1.04 is the growth factor applied each time period \(t\). The base 1.04 represents a 4% increase per year.
Read the Math card aloud. Have students practice saying "f of t equals 30,000 times 1.04 to the power of t." Emphasize: "to the power of" is the standard way to read an exponent. Ask a student to explain the Meaning card in their own words.
Math As A Second Language
General Form of an Exponential Function
Math · We Say · Meaning
★ Math (given)
\(a \cdot b^t\)
▲ We Say
“a times b to the power of t”
● Meaning
The general form of an exponential function: \(a\) is the initial value and \(b\) is the constant growth or decay factor applied over each equal interval \(t\).
Connect this general form back to the tuition example: a = 30,000, b = 1.04. Ask: "If b is greater than 1, is the function growing or decaying?" (Growing.) "If b is between 0 and 1?" (Decaying.) This previews Activity 2.
Math As A Second Language
Negative Exponent in Context
Math · We Say · Meaning
★ Math (given)
\(b^{-t}\)
▲ We Say
“b to the negative t”
● Meaning
A negative exponent represents evaluating an exponential function before the starting point. Example: \((1.04)^{-5}\) gives the value 5 years before the reference year.
Reinforce: "negative t" does not mean the value is negative — it means we are looking backward in time from the reference year. Have students say "b to the negative t" aloud.
Activity 2 · Launch ⏱ 3 min
Two Vans and Their Values
Clarify the meaning of depreciation for students who may not know what it means: when the value of an item depreciates, it means the item loses value over time, usually due to wear and tear from using it. Arrange students in groups of 2. Give them a moment to think quietly about the first question (which graph?), and then share their response with a partner. They should be ready to explain how they know certain graphs cannot represent the function.
A small business bought a van for $40,000. The van depreciates by 15% every year after its purchase. Depreciation means the item loses value over time.
⏱ 1 min individual think time before partner work
Activity 2 · Work Time ⏱ 12 min
Two Vans and Their Values
🧑🏫 Monitor for: students who identify Graph B by checking the y-intercept (40,000) and the value after 1 year (34,000 = 85% of 40,000). Look for students who write an expression using \(40{,}000 \cdot (0.85)^t\). Common error: students may confuse 15% depreciation with multiplying by 0.15 instead of 0.85.
If Students Are Stuck
If students are unsure how to start calculating the value of the van after 8 years, consider asking: "How did you decide which graph correctly represents the value of the van?" "How could you use a table to find the value of the car after 8 years?"
Amplify · Activity Builder · Type-in
Setup: Text + Math Response (answer) + Note with CL
Problem 1: Which graph?
content:
when answer.submitted and answer.latex = "B"
"Correct! Graph B shows exponential decay starting at 40,000."
when answer.submitted and answer.latex = "b"
"Correct! Graph B shows exponential decay starting at 40,000."
when answer.submitted and answer.latex = "A"
"Not quite — Graph A shows the value increasing. Does a depreciating van gain value?"
when answer.submitted and answer.latex = "C"
"You're close — Graph C shows decay, but it halves each year (factor of 0.5). A 15% decrease means the van retains 85%, not 50%."
when answer.submitted and answer.latex = "D"
"Good start — Graph D shows a decrease, but it is a straight line (linear). Depreciation by a percentage is not linear."
when answer.submitted
"Check: the van starts at $40,000 and loses 15% each year. After 1 year, it is worth 40000 times 0.85 = 34,000. Which graph passes through (0, 40000) and (1, 34000)?"
otherwise ""
Problem 2: Value after 8 years
content:
when answer.submitted and answer.latex = "10899.62"
"Correct! 40000(0.85)^8 is approximately $10,899.62."
when answer.submitted and answer.latex = "10,899.62"
"Correct! 40000(0.85)^8 is approximately $10,899.62."
when answer.submitted and answer.latex = "10899.6"
"Correct! 40000(0.85)^8 is approximately $10,899.62."
when answer.submitted
"Write the expression 40000 times (0.85)^8 and evaluate with a calculator."
otherwise ""
Problem 3: Car vs van after 8 years
content:
when answer.submitted
"Look for: students state the car is worth more. 30000(0.90)^8 is approximately $12,914.06, which is more than $10,899.62."
otherwise ""
Graph A
Graph B
Graph C
Graph D
Work with your partner
Which graph represents the value of the van over time? Explain how you know the other graphs do not fit.
Find the value of the van 8 years after its purchase. Show your reasoning.
The business also bought a car that cost $10,000 less than the van and depreciates at 10% per year. Would the car be worth more or less than the van 8 years after the purchase? Explain or show your reasoning.
On the same coordinate plane as the graph you chose, sketch a graph that shows the value of the car.
Try Saying
“I chose Graph ___ because ___.”
“Graph ___ does not fit because ___.”
Activity 2 · Synthesis ⏱ 5 min
Connecting Percentage Change to Exponential Functions
Display 2–3 strategies from previously selected students. Goal: students connect a constant percentage change to exponential decay. Make sure students see that: (1) the $40,000 price appears as the vertical intercept; (2) the 15% decay rate means the van retains 85% each year — \(40{,}000 \times 0.85 = 34{,}000\), so (1, 34000) must be on the graph; (3) to find the value after 8 years, write \(40{,}000 \cdot (0.85)^8\); (4) for the car, \(30{,}000 \cdot (0.90)^8 \approx 12{,}914\).
The goal of this discussion is to connect a constant percentage change to exponential growth or decay and see how initial value and decay factor show up in each representation.
“What do the strategies have in common? How are they different?”
“How does the initial value of the van show up in each method?”The $40,000 price of the van appears as the vertical intercept of the graph and as the coefficient in the equation \(40{,}000 \cdot (0.85)^t\).
“How does the percentage decrease show up in each method?”The 15% rate of decay tells you that after 1 year, the van retains 85% of its value. In the equation, the base is 0.85. On the graph, the point \((1, 34{,}000)\) must be on the correct curve.
Calculating the decay factor means subtracting the depreciation from 100% to find the remaining value. The van loses 15% each year, so each year it is worth 85% of the previous year's amount. For the car: it loses 10%, so it retains 90% — the decay factor is 0.90.
To find the value after 8 years, write an expression for the value \(t\) years after purchase: \(40{,}000 \cdot (0.85)^t\) for the van and \(30{,}000 \cdot (0.90)^t\) for the car. Evaluate at \(t = 8\).
click to advance discussion ▶
Try Saying
“I notice that the decay factor is ___ because ___.”
“The initial value shows up as ___ in the graph and as ___ in the equation.”
Math As A Second Language
Exponential Decay Function
Math · We Say · Meaning
★ Math (given)
\(40{,}000 \cdot (0.85)^t\)
▲ We Say
“40,000 times 0.85 to the power of t”
● Meaning
An exponential decay function where 0.85 means the quantity retains 85% of its value each period, reflecting a 15% decrease per year.
Connect to the previous MASL card for \(a \cdot b^t\): here \(a = 40{,}000\) and \(b = 0.85\). Ask: "How do we know this is decay and not growth?" (The base, 0.85, is between 0 and 1.) Have students say the We Say card aloud.
In the week that ended on January 4, there were 24 people who knew an exciting piece of news. In the 5 weeks that followed, the number of new people who learn the news grows exponentially.
On your own
Which expression represents the number of people who know the news \(w\) weeks after January 4? Explain what each number means.
(A) \(24 \cdot (1.5)^w\) (B) \(24 \cdot (0.5)^w\) (C) \(1.5 \cdot (24)^w\) (D) \(24 + 1.5w\)
If the trend continued, would the number of people who know the news be greater than 300 after 5 weeks? Explain or show how you know.
🧑🏫 Leave this slide on screen during the cool-down. The How To table is a reference for evaluating exponential expressions. Look for: students who correctly identify (A) and explain 24 as the initial value and 1.5 as the growth factor. Common error on problem 2: students may multiply 24 × 5 × 1.5 instead of computing \(24 \cdot (1.5)^5\).
Amplify · Activity Builder · Cool-Down
Setup: Text + Math Response (answer) + Note with CL
Input format example: Example: A
Problem 1: Which expression?
content:
when answer.submitted and answer.latex = "A"
"Correct! 24 is the initial number and 1.5 is the weekly growth factor."
when answer.submitted and answer.latex = "a"
"Correct! 24 is the initial number and 1.5 is the weekly growth factor."
when answer.submitted and answer.latex = "B"
"Not quite — 0.5 would mean the number of people is halving each week. Is the news spreading or shrinking?"
when answer.submitted and answer.latex = "C"
"You're close — but the initial value should be 24, not the base. Which part of a times b^t is the initial value?"
when answer.submitted and answer.latex = "D"
"Almost! 24 + 1.5w is a linear expression, not exponential. Exponential growth means multiplying by the same factor each week."
when answer.submitted
"The general form is a times b^t. The initial number of people is 24 and the number grows each week. Which expression fits?"
otherwise ""
Problem 2: Greater than 300 after 5 weeks?
content:
when answer.submitted and answer.latex = "No"
"Correct! 24(1.5)^5 = 24 times 7.59375 = 182.25, which is less than 300."
when answer.submitted and answer.latex = "no"
"Correct! 24(1.5)^5 = 24 times 7.59375 = 182.25, which is less than 300."
when answer.submitted and answer.latex = "Yes"
"Not quite — try computing 24 times (1.5)^5. Is the result above or below 300?"
when answer.submitted and answer.latex = "yes"
"Not quite — try computing 24 times (1.5)^5. Is the result above or below 300?"
when answer.submitted
"Substitute w = 5 into 24(1.5)^w. Evaluate (1.5)^5 first, then multiply by 24. Compare to 300."
otherwise ""
Lesson Synthesis ⏱ 5 min
Equal Factors over Equal Intervals
Use a new example to tie everything together: "A car costs $20,000 and loses 20% of its value every year." Walk through how the equal factor of 0.8 shows up in: (1) the description, (2) a table, (3) a graph, and (4) the equation \(f(t) = 20{,}000 \cdot (0.8)^t\). Consider drawing arrows from one row to the next in the table, showing multiplication by 0.8.
Exponential functions change by equal factors over equal intervals. The equal factor can be observed in every representation of the function.
Consider this situation: "A car costs $20,000 and loses 20% of its value every year." How does the equal factor show up in each representation?
In the description: Losing 20% of its value every year means that every year the value is multiplied by the same factor, \(0.8\).
In a table (and a graph): Each entry is \(0.8\) of the previous entry. You can draw arrows from one row to the next showing multiplication by \(0.8\).
In the equation: \(f(t) = 20{,}000 \cdot (0.8)^t\). The initial value is $20,000. The common factor by which the value changes each year is \(0.8\).
“How do you know whether a function represents growth or decay just by looking at the equation?”If the base \(b\) is greater than 1, the function is growing. If \(b\) is between 0 and 1, the function is decaying.
click to advance discussion ▶
Try Saying
“I know the function is growth/decay because the base is ___, which is ___.”
“The equal factor shows up in the table as ___ and in the equation as ___.”
By the end of this lesson, you can:
Explain (in writing) the meaning of different constants in an expression representing an exponential function.
Interpret equations and graphs that represent exponential functions in context.
Understand that exponential functions change by equal factors over equal intervals.
