Exponential Functions and Graphs
📊 Part 1: Understanding Exponential Functions
1. Circle the correct definition of an exponential function:
A function where the variable is in the base: y = x²
A function where the variable is in the exponent: y = 2ˣ
A function that increases by the same amount each time: y = 2x + 3
2. Which of the following are exponential functions? Check all that apply:
y = 3ˣ
y = x³
y = (1/2)ˣ
y = 5x + 2
y = 4ˣ⁺¹
3. Fill in the blanks:
In the exponential function y = 2ˣ, the number _____ is called the ________, and the variable _____ is called the ________.
📋 Part 2: Creating Tables of Values
4. Complete the table of values for y = 3ˣ:
5. Complete the table of values for y = (1/2)ˣ:
6. Looking at your tables above, which function shows exponential growth and which shows exponential decay?
Growth: _____________ Decay: _____________
7. Explain in your own words the difference between exponential growth and exponential decay:
📈 Part 3: Graphing Exponential Functions
8. Using your table from question 4, plot the graph of y = 3ˣ on the grid below:
9. Using your table from question 5, plot the graph of y = (1/2)ˣ on the grid below:
10. Circle the key features that ALL exponential graphs have:
y-intercept at (0, 1)
x-intercept at (1, 0)
Horizontal asymptote at y = 0
Passes through the origin
Always increasing
🌍 Part 4: Real-World Applications
11. Match each real-world situation with the type of exponential function it represents:
1. Population of rabbits doubling each year
2. Value of a car decreasing by 20% each year
3. Money in a savings account with compound interest
4. Radioactive material decaying over time
A. Exponential decay
B. Exponential growth
C. Exponential decay
D. Exponential growth
12. A bacteria culture starts with 100 bacteria and doubles every hour. The function is B(t) = 100 × 2ᵗ, where t is time in hours.
a) How many bacteria will there be after 3 hours? ____________
b) How many bacteria will there be after 5 hours? ____________
c) Is this exponential growth or decay? ____________
13. Sarah invests $1000 in a savings account that earns 5% compound interest annually. Write an exponential function to represent this situation:
14. Give two more examples of real-world situations that could be modelled using exponential functions:
🎯 Part 5: Challenge Questions
15. Without calculating, predict which function will have larger values when x = 10:
y = 2ˣ
y = 3ˣ
They will be equal
16. Explain your reasoning for question 15:
17. The half-life of a radioactive substance is 50 years. If you start with 800 grams, write the exponential decay function:
18. Sketch what you think the graph of y = 4ˣ would look like compared to y = 2ˣ:
💭 Part 6: Reflection
19. What is one key difference between exponential growth and linear growth?
20. Name one thing you found challenging about exponential functions and one thing you found interesting:
Challenging: ________________________________________________
Interesting: ________________________________________________
