Graphing Exponential Functions

Active learning helps students grasp the rapid, non-linear nature of exponential growth, which simple lectures often miss. Simulations and debates make compound interest visible and memorable, turning abstract formulas into concrete financial decisions students will face soon.

• 9th Grade
• Mathematics
• 3 activities
• 20 min–45 min

Learning Objectives

1. Analyze the behavior of exponential graphs, identifying the horizontal asymptote and y-intercept.
2. Compare the steepness of exponential curves based on different base values.
3. Explain the meaning of the y-intercept in the context of exponential growth and decay models.
4. Calculate the value of an exponential function at specific points given its equation.
5. Justify why an exponential graph approaches but never touches its horizontal asymptote.

Simulation Game

⏱ 45 min·Small Groups

Simulation Game: The Millionaire's Club

Groups are given a fictional $1,000 and three different 'investment' options with different interest rates and compounding periods. They must use the formula to calculate their balance after 10, 20, and 40 years, discovering the massive impact of time on their wealth.

Prepare & details

Justify why an exponential graph never crosses its horizontal asymptote.

Facilitation Tip: During 'The Millionaire's Club,' circulate and ask guiding questions like, 'What happens to your balance between compounding periods?' to focus attention on the mechanics of compounding.

Setup: Flexible space for group stations

Materials: Role cards with goals/resources, Game currency or tokens, Round tracker

• Apply
• Analyze
• Evaluate
• Create
• Social Awareness
• Decision-Making

Think-Pair-Share

⏱ 20 min·Pairs

Think-Pair-Share: Simple vs. Compound

One student calculates the interest on $500 at 10% for 5 years using simple interest (adding $50 each year). The other uses compound interest. They then compare their totals and discuss why the compound interest 'gap' gets wider every year.

Prepare & details

Analyze how the base of the function affects the steepness of the curve.

Facilitation Tip: In 'Simple vs. Compound,' provide colored pens so students can visually track how interest is added to the balance each period, reinforcing the idea of accelerating growth.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Formal Debate

⏱ 30 min·Small Groups

Formal Debate: Credit Card Caution

Students are shown how a small credit card balance can grow exponentially if only the minimum payment is made. They must debate the 'pros and cons' of using credit, using their mathematical models to prove how much 'extra' the item actually costs in the long run.

Prepare & details

Explain what the y-intercept represents in a growth or decay model.

Facilitation Tip: For the debate, assign roles clearly and give a two-minute warning before each speaker to keep the discussion focused and ensure all voices are heard.

Setup: Two teams facing each other, audience seating for the rest

Materials: Debate proposition card, Research brief for each side, Judging rubric for audience, Timer

• Analyze
• Evaluate
• Create
• Self-Management
• Decision-Making

Teaching This Topic

Teachers often begin by connecting exponential functions to students' immediate interests, like savings or loans, to build relevance. Emphasize the difference between simple and compound interest early, as this foundational concept prevents later confusion. Use real-world data, such as current interest rates, to make the lesson timely and meaningful, and avoid rushing to the formula before students see the pattern in the numbers.

What to Expect

Students will recognize how compounding frequency affects growth, compare simple and compound interest accurately, and explain why high interest with rare compounding can underperform lower interest with frequent compounding. They will also articulate the meaning of key features like the base and asymptote in context.

Watch Out for These Misconceptions

Common Misconception

During 'The Millionaire's Club,' watch for students assuming that the account with the highest interest rate will always end up with the most money, regardless of compounding frequency.

What to Teach Instead

Remind students to compare the final balances in the simulation, then guide them to notice how frequent compounding can overcome a slightly lower rate.

Common Misconception

During 'Simple vs. Compound,' watch for students calculating interest only on the original principal, ignoring the new total each period.

What to Teach Instead

Ask students to trace their calculations with a highlighter, showing how the 'new total' becomes the base for the next interest calculation.

Assessment Ideas

Quick Check

After 'The Millionaire's Club,' provide the equation y = 2 * (3)^x and ask students to identify the base, y-intercept, and horizontal asymptote, then sketch the graph labeling these features.

Discussion Prompt

During 'Simple vs. Compound,' ask students to explain how the graphs of two accounts with bases 1.05 and 1.10 (starting at $1000) will differ over 10 years, focusing on what the base represents in the growth formula.

Exit Ticket

After the 'Credit Card Caution' debate, give students a graph of an exponential decay function and ask them to write the equation of the horizontal asymptote and explain what it means in context, such as drug concentration in the bloodstream.

Extensions & Scaffolding

• Challenge: Ask students to research and compare the growth of two real savings accounts over 5 years, one with daily compounding and one with monthly, using actual bank rates.
• Scaffolding: Provide a partially completed table for 'Simple vs. Compound' where students fill in only the interest and new balance each year.
• Deeper: Have students model a scenario where they make regular monthly deposits into a compound interest account and graph the growth over time.

Key Vocabulary

Exponential Function	A function where the independent variable appears in the exponent, typically in the form y = a * b^x, where 'b' is the base.
Base (b)	The constant factor that is repeatedly multiplied in an exponential function. It determines the rate of growth or decay.
Horizontal Asymptote	A horizontal line that the graph of a function approaches as the input (x) tends towards positive or negative infinity.
Y-intercept	The point where the graph of a function crosses the y-axis, occurring when the input (x) is zero.
Exponential Growth	A pattern where a quantity increases by a constant multiplicative factor over equal intervals of time, resulting in a curve that rises steeply.
Exponential Decay	A pattern where a quantity decreases by a constant multiplicative factor over equal intervals of time, resulting in a curve that falls steeply.