Exponential functions

INTRODUCTION TO EXPONENTIAL FUNCTIONS

• Date: 3-9-26

• Student Name: Clae Hartman

• Period: 1

EXPONENTIAL FUNCTION AND GRAPH

• Example Functions:

  • Linear Function: $f(x) = 2x$

  • Exponential Function: $g(x) = 2^x$

Sketching Asymptotes

• Task: Sketch a line to represent the asymptote of the given exponential function and write the equation of the line.

• Suggest using a graphing calculator to assist in graphical representation.

COMPARISON OF FUNCTIONS: LINEAR vs EXPONENTIAL

• Commonality: Both functions have an x-related variable.

• Differences:

  • The linear function increases at a constant rate.

  • The exponential function increases at a rate proportional to its current value (growth rate increases over time).

  • An exponential function has a variable raised to an exponent (x).

TASK I: TABLES

• Objective: Complete the table of values for each function.

TASK II: GRAPHS

• Objective: Using the completed tables, create and label graphs for each function.

EXAMPLES OF VALUES

• Function Values:

  • **For g(x) = 4 - (0.25) **:

  • $g(x) = 4 - (0.25)^x$

TABLE OF VALUES

DIFFERENCES BETWEEN LINEAR AND EXPONENTIAL FUNCTIONS

• The exponential function has a characteristic growth rate that is not constant; it grows rapidly compared to the linear function.

TASKS RELATED TO OPTION PLANS FOR SAVINGS

SCENARIO

• Individual: Shawn

• Current Savings: $10

• Time Frame: Several months

OPTION 1

• Description: Deposits $20 each month into savings.

• Function Representation:

  • $y = 20x + 10$

  • Where $y$ is total amount in savings, and $x$ is number of months.

Table for OPTION 1

OPTION 2

• Description: Doubles the amount in savings each month.

• Function Representation:

  • $y = 10 imes 2^x$

  • Where $y$ is total amount in savings, and $x$ is number of months.

Table for OPTION 2

COMPARISON OF OPTIONS

6. Which option will result in higher savings?

   • Conclusion: Option 2 will result in Shawn saving more money due to exponential growth of savings compared to linear deposit.

SUMMARY OF DIFFERENCES BETWEEN OPTIONS

7. Main Differences:

• Option 1 (Linear Function):

  • Savings increase by a constant amount ($20) each month.

  • Resulting total savings grows gradually.

• Option 2 (Exponential Function):

  • Savings double each month starting from $10.

  • Results in significant growth in total savings over the same time frame due to the nature of exponential growth.

Exponential functions are a fundamental area of mathematics that model growth and decay. For example, the exponential function $g(x) = 2^x$ exhibits rapid growth as the value of $x$ increases, which is contrasted with linear functions that grow at a constant rate, such as $f(x) = 2x$. In a practical application, let's look at two savings options for an individual named Shawn who starts with $10.

OPTION 1: Linear Savings

• Deposits $20 each month. The function can be represented as:
  $y = 20x + 10$.
  This means for every month that passes, the total savings grow gradually by a constant amount, as seen in the savings table. After 6 months, Shawn will have $130 in savings.

OPTION 2: Exponential Savings

• Doubles his savings each month:
  $y = 10 imes 2^x$.
  By the end of 6 months, Shawn's savings will skyrocket to $640 due to the nature of exponential growth. Thus, it can be concluded that exponential functions lead to significantly larger increases in savings compared to linear functions, especially over time.

The episode can further explore the implications of these functions in real-world scenarios, along with the effects of compound interest and investment strategies that leverage exponential growth. Audience engagement can be increased by inviting listeners to share their experiences with savings and exponential growth.