Grade: Grade 11 Subject: Mathematics Unit: Functions & Modeling SAT: AdvancedMath ACT: Math

Guided Practice: Functions & Modeling

Learn

This guided practice lesson reinforces the key concepts from exponential, logarithmic, and real-world modeling. You will work through problems with detailed step-by-step guidance to build confidence before attempting independent practice.

Key Concepts Review

  • Exponential Functions: f(x) = a * b^x where a is the initial value and b is the growth/decay factor
  • Logarithmic Functions: The inverse of exponential functions; log_b(x) answers "what power of b gives x?"
  • Growth vs. Decay: Growth when b > 1; decay when 0 < b < 1
  • Continuous Compound Interest: A = Pe^(rt)
  • Half-life Formula: A = A_0 * (1/2)^(t/h) where h is the half-life

Problem-Solving Strategy

  1. Identify the type of function (exponential, logarithmic, or combination)
  2. Determine what quantity you need to find
  3. Set up the equation using the appropriate formula
  4. Solve using algebraic techniques or logarithms
  5. Check your answer in the original context

Worked Examples

Example 1: Exponential Growth

Problem: A population of bacteria doubles every 3 hours. If the initial population is 500, find the population after 12 hours.

Show Solution

Step 1: Identify the model type - This is exponential growth with doubling.

Step 2: Set up the formula: P(t) = P_0 * 2^(t/d) where d is doubling time

Step 3: Substitute values: P(12) = 500 * 2^(12/3) = 500 * 2^4

Step 4: Calculate: P(12) = 500 * 16 = 8,000 bacteria

Example 2: Solving with Logarithms

Problem: An investment of $2,000 grows at 5% annual interest compounded continuously. How long until it reaches $3,000?

Show Solution

Step 1: Use the continuous compound formula: A = Pe^(rt)

Step 2: Substitute: 3000 = 2000 * e^(0.05t)

Step 3: Divide: 1.5 = e^(0.05t)

Step 4: Take natural log: ln(1.5) = 0.05t

Step 5: Solve: t = ln(1.5)/0.05 = 0.405/0.05 = 8.11 years

Example 3: Logarithmic Equation

Problem: Solve for x: log_2(x + 3) + log_2(x - 1) = 5

Show Solution

Step 1: Use log property: log_2[(x+3)(x-1)] = 5

Step 2: Convert to exponential: (x+3)(x-1) = 2^5 = 32

Step 3: Expand: x^2 + 2x - 3 = 32

Step 4: Solve quadratic: x^2 + 2x - 35 = 0

Step 5: Factor: (x + 7)(x - 5) = 0, so x = -7 or x = 5

Step 6: Check domain: x must be > 1, so x = 5 only

Practice Problems

Work through these problems step-by-step. Solutions are provided for self-checking.

Problem 1: A radioactive substance has a half-life of 4 days. If you start with 80 grams, how much remains after 20 days?

Show Answer

A = 80 * (1/2)^(20/4) = 80 * (1/2)^5 = 80/32 = 2.5 grams

Problem 2: Solve: 3^(2x-1) = 27

Show Answer

3^(2x-1) = 3^3, so 2x - 1 = 3, therefore x = 2

Problem 3: A car depreciates at 15% per year. If it costs $25,000 new, what is its value after 6 years?

Show Answer

V = 25000 * (0.85)^6 = 25000 * 0.377 = $9,428.74

Problem 4: Evaluate: log_3(81) - log_3(9)

Show Answer

log_3(81/9) = log_3(9) = log_3(3^2) = 2

Problem 5: The population of a city is modeled by P(t) = 50000 * e^(0.02t). Find the population after 25 years.

Show Answer

P(25) = 50000 * e^(0.5) = 50000 * 1.649 = 82,436 people

Problem 6: Solve: ln(x) + ln(x - 2) = ln(15)

Show Answer

ln[x(x-2)] = ln(15), so x^2 - 2x = 15, x^2 - 2x - 15 = 0, (x-5)(x+3) = 0. Since x > 2, x = 5

Problem 7: How long does it take for an investment to triple at 8% annual interest compounded continuously?

Show Answer

3P = Pe^(0.08t), ln(3) = 0.08t, t = ln(3)/0.08 = 13.73 years

Problem 8: Write the equation for exponential decay if an initial amount of 1000 decreases by 25% each year.

Show Answer

A(t) = 1000 * (0.75)^t or equivalently A(t) = 1000 * e^(-0.288t)

Problem 9: Simplify: log_5(125) + 2*log_5(5) - log_5(25)

Show Answer

3 + 2(1) - 2 = 3

Problem 10: A medicine's concentration in the bloodstream decreases by 20% every hour. If the initial dose is 400 mg, when will the concentration fall below 50 mg?

Show Answer

50 = 400 * (0.8)^t, 0.125 = 0.8^t, ln(0.125) = t*ln(0.8), t = ln(0.125)/ln(0.8) = 9.32 hours

Check Your Understanding

Answer these questions to test your mastery of the guided practice concepts.

  1. What is the first step when solving an exponential equation where the bases are not the same?
  2. Why is it important to check the domain when solving logarithmic equations?
  3. How do you convert between exponential and logarithmic form?
  4. What does a negative exponent mean in the context of exponential decay?

Next Steps

  • If you found the problems manageable, proceed to the Word Problems lesson
  • If you struggled, review the Exponential and Logarithmic lesson first
  • Practice converting between exponential and logarithmic forms
  • Create your own problems using real-world contexts