Week 9, Session 18: Exponential Growth and Decay

Week 9, Session 18: Exponential Growth and Decay

Self-study purpose

Some GED® patterns grow or shrink by multiplication instead of addition. This session teaches how to recognize and use exponential models.

What you should be able to do after this session

• Distinguish linear from exponential change.
• Identify growth and decay factors.
• Evaluate simple exponential expressions.

Visual model

Core concept

Exponential patterns multiply by the same factor each step. Growth has a factor greater than 1. Decay has a factor between 0 and 1.

Deep self-study explanation

Exponential patterns change by repeated multiplication. This is different from linear change, which uses repeated addition. Doubling, tripling, halving, and percent increase or decrease are signal words for exponential models. The base tells the growth or decay factor, while the coefficient tells the initial value.

Step-by-step method

1. Check whether outputs add by the same amount or multiply by the same factor.
2. Identify the initial value at \(x=0\).
3. Identify the growth or decay factor.
4. Write a model such as \(y=a(b^x)\).
5. Evaluate powers before multiplying.

Guided problem-solving script

Check consecutive outputs by division. If the ratio is constant, think exponential. Identify the starting value at \(x=0\). Convert percent changes into factors: increase by r means 1+r, decrease by r means 1-r. Evaluate the exponent before multiplying by the starting value.

Worked GED®-style example

Problem: A population starts at 100 and doubles every hour. Write a model.

Step 1: Initial value is 100.
Step 2: Doubling means multiply by 2 each hour.
Step 3: Use \(P=100(2^t)\).
Answer: \(P=100(2^t)\).

Why each step works

The worked example turns 'starts at 100' into the coefficient and 'doubles' into the base 2. The exponent t counts how many times the doubling happens.

Ten guided examples with increasing difficulty

Example 1 - Foundation

Problem: A pattern is 3, 6, 12, 24. Is it linear or exponential?
Solution:

1. The values multiply by 2 each step.
2. Repeated multiplication means exponential.

Answer: Exponential.

Example 2 - GED®-level

Problem: A value starts at 80 and increases by 25% each year. Write a model.
Solution:

1. Initial value is 80.
2. A 25% increase means a factor of \(1.25\).
3. Use \(y=80(1.25^x)\).

Answer: \(y=80(1.25^x)\).

Example 3 - Challenge

Problem: A machine loses 18% of its value each year. It starts at $900. What is its value after 2 years?
Solution:

1. If 18% is lost, 82% remains, so the factor is 0.82.
2. Use \(V=900(0.82^t)\).
3. Substitute \(t=2\): \(V=900(0.82^2)\).
4. \(0.82^2=0.6724\), so \(V=900(0.6724)=605.16\).

Answer: About $605.16.

Example 4 - Extra foundation

Problem: Which table shows exponential growth?
Solution:

1. Exponential growth multiplies by the same factor.
2. 2 to 6 to 18 to 54 multiplies by 3.
3. So it is exponential.
4. Check the trap: Linear growth adds; exponential growth multiplies.

Answer: x: 0,1,2,3; y: 2,6,18,54

Example 5 - Extra skill builder

Problem: For \(y=3(2^x)\), what is the initial value at \(x=0\)?
Solution:

1. Substitute \(x=0\).
2. \(2^0=1\).
3. \(y=3(1)=3\).
4. Check the trap: Any nonzero base to the zero power equals 1.

Answer: \(3\)

Example 6 - Extra GED®-level

Problem: For \(y=5(0.8^x)\), the function shows:
Solution:

1. The base is 0.8.
2. A base between 0 and 1 means repeated multiplication by a fraction.
3. The outputs decrease, so it is decay.
4. Check the trap: A positive base below 1 means decay.

Answer: decay

Example 7 - Extra multi-step

Problem: A population doubles every year from 100. Which model fits?
Solution:

1. Initial value is 100.
2. Doubling means multiply by 2 each year.
3. So \(P=100(2^t)\).
4. Check the trap: Doubling is multiplicative, not additive.

Answer: \(P=100(2^t)\)

Example 8 - Extra reasoning

Problem: A value decreases by 20% each step. What is the decay factor?
Solution:

1. Keep 100%-20%=80%.
2. Convert 80% to decimal 0.80.
3. The factor is 0.80.
4. Check the trap: The decay factor is what remains, not what is lost.

Answer: \(0.80\)

Example 9 - Extra challenge

Problem: A value increases by 15% each step. What is the growth factor?
Solution:

1. Start with 100%.
2. Add 15% to get 115%.
3. Convert to 1.15.
4. Check the trap: Growth factors are greater than 1.

Answer: \(1.15\)

Example 10 - Extra mastery

Problem: For \(y=4(3^x)\), what is y when \(x=2\)?
Solution:

1. Substitute \(x=2\).
2. \(3^2=9\).
3. \(4(9)=36\).
4. Check the trap: Evaluate the exponent before multiplying.

Answer: \(36\)

Common mistakes to avoid

• Treating doubling as adding 2.
• Using the percent lost instead of the percent remaining.
• Forgetting that \(b^0=1\).

Mastery routine

Write one growth model and one decay model from real examples. Label the initial value and factor.

Practice ladder for independent study

1. Decide whether patterns add or multiply.
2. Find common ratios.
3. Convert percent growth to factors.
4. Convert percent decay to factors.
5. Write and evaluate exponential models.

Correction checklist

• Did I use multiplication instead of addition?
• Did I identify the initial value?
• Did I use the percent remaining for decay?
• Did I evaluate powers before multiplying?
• Does the model grow or decay as expected?