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Uncovering Deceptive Growth
Algebra 2 // Unit: Exponentials
Mission Briefing
Objective
Analyze exponential functions with manipulated exponents to find the effective growth rate.
"Higher growth factors will always grow faster than smaller growth factors—once you find the true base."
Agenda
1 Power of a Power Warm-up
2 Video: Comparing Exponentials
3 Activity: Deceptive Rates
4 Reflection Journal
Warm-up: Power of a Power
Recall the Rule:
\( (a^m)^n = a^{m \cdot n} \)
Simplify:
\( (1.05^2)^x \)
Rewrite as \( (b^k)^x \):
\( 1.25^{x/2} \)
Analyzing the Trick
Embedded media
Watching from 4:17 to the End
Get ready to pause at 5:00!
The Deception at 5:00
Function C looks like this:
\( f(x) = 400(1.25)^{x/2} \)
Many people see 1.25 and think "25% growth."
But that's not the growth factor per 1 unit of \( x \)!
The Solution
Rewrite the base:
\( (1.25^{1/2})^x \)
\( (\sqrt{1.25})^x \)
\( (1.118)^x \)
Actual growth rate: 11.8%
DECEPTIVE RATES
Your task: Take the list of functions on your worksheet, find their effective base , and rank them.
20
Minutes
6
Functions
Reflection Journal
Investigator's Report
"How can advertisers or banks use non-standard time periods (like exponents of \( 2x \) or \( x/4 \)) to make an interest rate look better or worse than it actually is?"
Provide one specific numerical example in your response.
Deceptive Rates Worksheet Rate Radar: Deceptive Rates
Algebra 2 // Exponential Analysis
Name:
Date:
PART 1
Exponent Rule Refresher
Rule: Power of a Power
\( (a^m)^n = a^{m \cdot n} \)
Use this to rewrite functions into standard form \( y = a(b)^x \).
Simplify:
\( (1.12^3)^x = \)
Rewrite as \( (b^k)^x \):
\( 1.05^{4x} = \)
PART 2
Video Intelligence: Pause @ 5:00
The Case of Function C: \( f(x) = 400(1.25)^{x/2} \)
Why is the growth factor NOT 1.25?
Show the steps to find the "True Base" (Effective growth factor):
PART 3
The Mission: Ranking Deception
Rewrite each function in standard form \( y = a(b)^x \). Then, identify the Effective Growth Factor (the value of \( b \)) and rank the functions from 1 (Slowest Growth) to 6 (Fastest Growth).
Function Rewritten Form Effective Base (\( b \)) Rank \( f(x) = 100(1.08)^{3x} \) \( g(x) = 50(2)^{x/4} \) \( h(x) = 20(3)^{x/2} \) \( j(x) = 500(1.5)^{2x} \) \( k(x) = 10(4)^{x/3} \) \( l(x) = 100(1.1)^{12x} \)
Pro Tip: Evaluate your "Effective Base" to at least 3 decimal places to ensure your ranking is precise!
Deceptive Rates Answer Key Deceptive Rates: Answer Key
Teacher Resource // 10th Grade Algebra 2
Part 1: Warm-up
Simplify
\( (1.12^3)^x = \mathbf{1.405^x} \)
Rewrite
\( 1.05^{4x} = \mathbf{(1.05^4)^x} \approx \mathbf{1.216^x} \)
Part 2: Video Analysis (Pause @ 5:00)
Why is the growth factor NOT 1.25?
The growth of 25% occurs only every 2 units of time (exponent is \( x/2 \)). To find the growth per 1 unit of time, we must evaluate the half-power.
Steps to find True Base:
\( 1.25^{x/2} = (1.25^{1/2})^x = \sqrt{1.25}^x \approx \mathbf{1.118}^x \)
Part 3: Ranking Deception
Function Rewritten Form Effective Base (\( b \)) Rank \( f(x) = 100(1.08)^{3x} \) \( (1.08^3)^x \) \( 1.2597... \approx \mathbf{1.260} \) 3 \( g(x) = 50(2)^{x/4} \) \( (2^{1/4})^x \) \( \sqrt[4]{2} \approx \mathbf{1.189} \) 2 \( h(x) = 20(3)^{x/2} \) \( (3^{1/2})^x \) \( \sqrt{3} \approx \mathbf{1.732} \) 5 \( j(x) = 500(1.5)^{2x} \) \( (1.5^2)^x \) \( 2.25 \) 6 (Fastest) \( k(x) = 10(4)^{x/3} \) \( (4^{1/3})^x \) \( \sqrt[3]{4} \approx \mathbf{1.587} \) 4 \( l(x) = 100(1.1)^{12x} \) \( (1.1^{12})^x \) \( 3.1384... \approx \mathbf{3.138} \) 7 (Super-fast) Reference C \( (1.25^{1/2})^x \) \( 1.118 \) 1 (Slowest)
Teaching Tip:
Wait, which is fastest? Even though Function L has the smallest base (1.1), the compounding frequency of 12 times per unit makes it the absolute fastest. Remind students that the visible growth factor (1.1) is deceptive without looking at the exponent. *Note: Rank 7 added to account for the Video Reference C being slowest overall in the context of all problems.*
Growth Reflection Journal Document Rate Radar
Investigation Log // 10th Algebra 2
Investigator:
Date:
Prompt #01: The Deception of Time
"Advertisers, credit card companies, and banks often describe interest rates using non-standard time periods. How can manipulating an exponent (like using \( 2x \) vs \( x/4 \)) make a rate look better or worse than it actually is for the consumer?"
Journal Entry
Supporting Evidence (Mathematical Example)
Unit: Exponential Functions Confidential // Mission DECEPTIVE RATES
